System and method for reducing pilot signal contamination using orthogonal pilot signals

ABSTRACT

A system for transmitting pilot signals has first signal processing circuitry for generating a pilot signal at a transmitting unit. An MLO modulation circuit modulates the pilot signal using quantum level overlay modulation to apply at least one orthogonal function to the pilot signal. A transceiver transmits the modulated pilot signal from the transmitting unit over a pilot channel. The at least one orthogonal function applied to the pilot signal substantially reduces pilot channel contamination on the pilot channel from other pilot channels.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application No.62/490,138, filed on Apr. 26, 2017 and entitled PRODUCT PATENT REDUCINGPILOT CONTAMINATION USING NEW ORTHOGONAL PILOT SIGNALS THAT REDUCETIME-BANDWIDTH RESOURCES, which is incorporated herein by reference inits entirety. This application is also a Continuation-in-Part of U.S.patent application Ser. No. 15/216,474, filed on Jul. 21, 2016 andentitled SYSTEM AND METHOD FOR COMBINING MIMO AND MODE DIVISIONMULTIPLEXING, which is incorporated herein by reference in its entirety.

TECHNICAL FIELD

The present system relates to pilot signal transmissions, and moreparticularly, to the use of orthogonal pilot signals for pilot signaltransmissions to reduce pilot signal contamination.

BACKGROUND

The use of voice and data networks has greatly increased as the numberof personal computing and communication devices, such as laptopcomputers, mobile telephones, Smartphones, tablets, et cetera, hasgrown. The astronomically increasing number of personal mobilecommunication devices has concurrently increased the amount of databeing transmitted over the networks providing infrastructure for thesemobile communication devices. As these mobile communication devicesbecome more ubiquitous in business and personal lifestyles, theabilities of these networks to support all of the new users and userdevices has been strained. Thus, a major concern of networkinfrastructure providers is the ability to increase their bandwidth inorder to support the greater load of voice and data communications andparticularly video that are occurring. Traditional manners forincreasing the bandwidth in such systems have involved increasing thenumber of channels so that a greater number of communications may betransmitted, or increasing the speed at which information is transmittedover existing channels in order to provide greater throughput levelsover the existing channel resources.

Transmitting devices transmit a pilot signal over a pilot channel to areceiving device in order to determine channel state information for acommunications channel between the transmitter and the receiver. Whenmultiple pilot channel signals are being transmitted from a number oftransmitting devices to a receiving device, interference between thepilot channels may cause pilot channel contamination. Thus, some mannerfor mitigating the effects of pilot channel contamination would greatlybenefit the communications process.

SUMMARY

The present invention, as disclosed and described herein, in one aspectthereof provides a system for transmitting pilot signals has firstsignal processing circuitry for generating a pilot signal at atransmitting unit. An MLO modulation circuit modulates the pilot signalusing quantum level overlay modulation to apply at least one orthogonalfunction to the pilot signal. A transceiver transmits the modulatedpilot signal from the transmitting unit over a pilot channel. The atleast one orthogonal function applied to the pilot signal substantiallyreduces pilot channel contamination on the pilot channel from otherpilot channels.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding, reference is now made to thefollowing description taken in conjunction with the accompanyingDrawings in which:

FIG. 1 illustrates pilot signal transmissions between a user equipment(UE) and a base station (BS);

FIG. 2 illustrates conditions for pilot channel contamination;

FIG. 3 illustrates a massive MIMO communications system;

FIG. 4 illustrates the use of multilevel overlay modulation with amassive MIMO system to reduce pilot channel communication;

FIG. 5 illustrates the transmission of a pilot channel andcommunications between a transmitter and a receiver;

FIG. 6 is a flow diagram illustrating the use of pilot channels toobtain channel state information;

FIG. 7 illustrates a system using MLO/QLO for transmissions between userdevices and a base station using a MIMO system;

FIG. 8 is a flow diagram illustrating the process of providing pilotchannel communications using the system of FIG. 7;

FIG. 9 illustrates a single input, single output (SISO) channel;

FIG. 10 illustrates a multiple input, multiple output (MIMO) channel;

FIG. 11 illustrates the manner in which a MIMO channel increasescapacity without increasing power;

FIG. 12 compares capacity between a MIMO system and a single channelsystem;

FIG. 13 illustrates multiple links provided by a MIMO system;

FIG. 14 illustrates various types of channels between a transmitter anda receiver;

FIG. 15 illustrates an SISO system, MIMO diversity system and the MIMOmultiplexing system;

FIG. 16 illustrates the loss coefficients of a 2×2 MIMO channel overtime;

FIG. 17 illustrates the manner in which the bit error rate declines as afunction of the exponent of the signal-to-noise ratio;

FIG. 18 illustrates diversity gains in a fading channel;

FIG. 19 illustrates a model decomposition of a MIMO channel with fullCSI;

FIG. 20 illustrates SVD decomposition of a matrix channel into parallelequivalent channels;

FIG. 21 illustrates a system channel model;

FIG. 22 illustrates the receive antenna distance versus correlation;

FIG. 23 illustrates the manner in which correlation reduces capacity infrequency selective channels;

FIG. 24 illustrates the manner in which channel information varies withfrequency in a frequency selective channel;

FIG. 25 illustrates antenna placement in a MIMO system;

FIG. 26 illustrates multiple communication links at a MIMO receiver;

FIG. 27 illustrates various techniques for increasing spectralefficiency within a transmitted pilot signal;

FIG. 28 illustrates a multiple level overlay transmitter system;

FIG. 29 illustrates an FPGA board;

FIG. 30 illustrates a multiple level overlay receiver system;

FIGS. 31A-31J illustrate representative multiple level overlay signalsand their respective spectral power densities;

FIG. 32 is a block diagram of a transmitter subsystem for use withmultiple level overlay;

FIG. 33 is a block diagram of a receiver subsystem using multiple leveloverlay;

FIG. 34 illustrates an equivalent discreet time orthogonal channel ofmodified multiple level overlay;

FIG. 35 illustrates the PSDs of multiple layer overlay, modifiedmultiple layer overlay and square root raised cosine;

FIG. 36 illustrates the various signals that that may be transmittedover different pilot channels from a transmitter to a receiver; and

FIG. 37 illustrates the overlapped absolute Fourier transforms ofseveral signals.

DETAILED DESCRIPTION

Referring now to the drawings, wherein like reference numbers are usedherein to designate like elements throughout, the various views andembodiments of a system and method for reducing pilot signalcontamination using orthogonal pilot signals are illustrated anddescribed, and other possible embodiments are described. The figures arenot necessarily drawn to scale, and in some instances the drawings havebeen exaggerated and/or simplified in places for illustrative purposesonly. One of ordinary skill in the art will appreciate the many possibleapplications and variations based on the following examples of possibleembodiments.

Massive MIMO has been recognized as a promising technology to meet thedemand for higher data capacity for mobile networks in 2020 and beyond.A Massive MIMO system includes multiple antennas transmitting betweentransmitting and receiving stations. In order to control communicationsover the multiple channels, channel state information (CSI) must beobtain concerning the communications channels. Referring now to FIG. 1,there is illustrated the manner in which a UE (user equipment) 102transmits a pilot channel to a baste station 104 in order to obtainchannel state information. The UE 102 transmits a pilot signal 106 tothe BS 104 so that the signal can be analyzed to obtain the CSI for anassociated channel. Each base station 104 within a Massive MIMO systemneeds an accurate estimation of the CSI on communications channels withuser equipment (UE), either through feedback or channel reciprocityschemes in order to achieve the benefits of massive MIMO.

Time division duplex (TDD) is one mode currently used to acquire timelyCSI in massive MIMO systems. The use of non-orthogonal pilot schemes,proposed for channel estimation in multi-cell TDD networks, isconsidered as a major source of pilot contamination due to thelimitations of coherence time. Referring now to FIG. 2, there isprovided an illustration of the conditions leading to pilotcontamination. Pilot contamination occurs when multiple UEs 202 aretransmitting multiple pilot signals 204 to a base station 206. A similarsituation can occur in a massive MIMO system where multiple pilotsignals 204 are being transmitted from multiple antennas to a receivinglocation. The multiple pilot signals 204 interfere with each othercausing pilot signal contamination that prevents accurate CSImeasurements. Other sources of pilot contamination include hardwareimpairment and non-reciprocal transceivers. Therefore any attempt to usebetter orthogonal pilot signals is critical for estimating the channelcorrectly and providing spectral efficiency gains via MIMO.

The increasing demand for higher data rates in wireless mobilecommunication systems, and the emergence of services like internet ofthings (IoT), machine-to-machine communication (M2M), e-health,e-learning and e-banking have introduced the need for new technologiesthat are capable of providing higher capacity compared to the existingcellular network technologies. It is projected that mobile traffic willincrease in the next decade in the order of thousands compared tocurrent demand; hence, the need for next generation networks that candeliver the expected capacity compared to existing network deployment.According to Cisco networking index, global mobile data traffic grew 69percent in 2014, making it 30 times the size of the entire globalinternet in 2000. The index also shows that wireless data explosion isreal and increasing at an exponential rate, which is driven largely bythe increased use of smart phones and tablets, and video streaming.

Key technology components that have been identified that requiresignificant advancement are radio links, multi-node/multi-antennatechnologies, multi-layer and multi-RAT networks, and spectrum usage. Inmulti-node/multi-antenna technologies, massive MIMO is being consideredin order to deliver very high data rates and spectral efficiency, aswell as enhanced link reliability, coverage and energy efficiency. Asshown generally in FIG. 3, massive MIMO, is a communication system wherea base station (BS) 302 with a few hundred antennas in an array 304simultaneously serve many tens of user terminals (UTs) 306, each havinga single antenna 308, in the same time-frequency resource.

A massive MIMO system as will be more fully described herein below usesantenna arrays for transmission between transceiving locations. Thebasic advantages offered by the features of massive MIMO can besummarized as follows:

Multiplexing gain: Aggressive spatial multiplexing used in massive MIMOmakes it theoretically possible to increase the capacity by 10× or more.

Energy efficiency: The large antenna arrays can potentially reduceuplink (UL) and downlink (DL) transmit powers through coherent combiningand an increased antenna aperture. It offers increased energy efficiencyin which UL transmit power of each UT can be reduced inverselyproportional to the number of antennas at the BS with no reduction inperformance.Spectral efficiency: The large number of service antennas in massiveMIMO systems and multiplexing to many users provides the benefit ofspectral efficiency.Increased robustness and reliability: The large number of antennasallows for more diversity gains that the propagation channel canprovide. This in turn leads to better performance in terms of data rateor link reliability. When the number of antennas increases withoutbound, uncorrelated noise, fast fading, and intra-cell interferencevanish.Simple linear processing: Because BS station antenna is much larger thanthe UT antenna (M>>K), simplest linear pre-coders and detectors areoptimal.Cost reduction in RF power components: Due to the reduction in energyconsumption, the large array of antennas allows for use of low cost RFamplifiers in the milli-watt range.

There are real world challenges such as channel estimation and pilotdesign, antenna calibration, link adaptation and propagation effects inmassive MIMO system. To achieve the benefits of massive MIMO inpractice, each BS 302 needs accurate estimation of the channel stateinformation (CSI), either through feedback or channel reciprocityschemes. There are different flavors of massive MIMO includingfrequency-division duplex (FDD) and time-division duplex (TDD). TDD isconsidered a better mode to acquire timely CSI over FDD because TDDrequires estimation, which can be done in one direction and used in bothdirections; while FDD requires estimation and feed-back for both forwardand reverse directions, respectively.

In TDD, the use of channel reciprocity and training signals (pilot) inthe UL are key features for its application. Using reciprocity, it isassumed that the forward channel is equal to the transpose of thereverse channel for mathematical analysis and simulations. Therefore,the required channel information is obtained from transmitted pilots onthe reverse link from UTs 306. However, in practice, an antennacalibration scheme must be implemented at the transmitter side and/orthe receiver sides owing to the different characteristics of transmit orreceive RF chains.

Pilot Contamination in Massive MIMO

Some have suggested the minimum number of UL pilot symbols may equal tothe number of UTs while others have shown that optimal number oftraining symbols can be larger than the number of antennas if trainingand data power are required to be equal. In most studies on pilotcontamination, it is assumed that the same size of pilot signals is usedin all cells. Contrary to this assumption, the studies have shown thatarbitrary pilot allocation is possible in multi-cell systems. Betterspectral efficiency in wireless networks requires appropriate frequencyor time or pilot reuse factors in order to maximize system throughput.The reuse of frequency has been shown to provide more efficient use ofthe limited available spectrum, but it also introduces co-channelinterference in a massive MIMO system. Therefore, both orthogonality aswell as efficient use of time-frequency resources is needed. These areprovided by QLO signals where they minimize time-bandwidth products andyet all signals are mutually orthogonal to one another.

The pilot signals which are used to estimate the channels can becontaminated as a result of reuse of non-orthogonal pilot signals in amulti-cell system. This phenomenon causes the inter-cell interferencethat is proportional to the number of BS antennas, which in turn reducesthe achievable rates in the network and affect the spectrum efficiency.Therfore, QLO pilot signals can resolve these degradations. There areseveral techniques on eliminating inter-cell interference in multi-cellsystems in which it is assumed that the BSs 302 are aware of CSI. Forinstance, coordinated beamforming have been proposed in multi-cellmulti-antenna wireless systems in eliminating inter-cell interferencewith the assumption that the CSI of each UT 306 is available at the BS302. However, QLO signals as pilots are very usefull in conjuction withthese techniques. In practical implementation, estimation of channelstate information is required. In the asymptotic regime, where the BS302 has an unlimited number of antennas and there is no cooperation inthe cellular network, not all interference vanishes because of reuse oforthogonal training sequences across adjacent cells leading tointer-cell interference. Therefore, QLO signals as pilots are necessaryto reduce such interference.

There are several techniques on reduction of inter-cell interferencewith a focus on mitigation of pilot contamination in channel estimation.Although most techniques have focused on the reuse of non-orthogonaltraining sequence as the only source of pilot contamination, there areother sources of pilot contamination. Other sources of pilotcontamination could be hardware impairments due to in-band andout-of-bound distortions that interfere with training signals andnon-reciprocal transceivers due to internal clock structures of RFchain. Therefore, QLO signals as pilots are critical.

Channel State Information (CSI)

Referring now back to FIG. 3, acquisition of timely and accurate CSI atthe BS 302 is very important in a wireless communication system and isthe central activity of massive MIMO. Good CSI helps to maximize networkthroughput by focusing of transmit power on the DL and collection ofreceive power on the UL via a selective process. Therefore, the need foran effective and efficient method for channel estimation (CE) iscritical. Channel state estimation error affects MIMO system performanceand the effect of imperfect channel knowledge has major implications tothe massive MIMO network. The estimation of the CSI can be driven bytraining sequences (pilot), semi-blind or blind, based techniques.Training overheads for CSI and CSI feedback contributes to increasedcost of CSI estimation and decreases multiple access channel efficiency.Here, we purpose different training methods with a focus on trainingsequence and semi-blind schemes which are based on the use of specialtype of pilots called QLO pilots that are both applicable to the CEoperations in multi-cell massive MIMO systems under the TDD and FDDschemes.

Training Methods

Training-based (TB) channel estimation is one manner for determining CSIin MIMO systems. In training-based estimation, pilot sequences known tothe receiver are transmitted over the channel. These known pilotsequences are used by the receiver to build estimates of the random MIMOchannel. Two different training schemes are being developed, theconventional time-multiplexed pilot scheme (CP) and superimposed pilotscheme (SIP). In CP, the pilot symbols are transmitted exclusively indedicated time slots allocated for training. In SIP, the pilot symbolsare superimposed to the data and data are transmitted in all time slots.QLO pilot signals can be applied to both techniques.

The performance analysis based on the maximum data rate for the CP andSIP scheme using different scenarios can be done to see which is betterfor spectral efficiency in different fading environments. Some haveconsidered different training schemes for both the flat-fading andfrequency-selective MIMO cases which include design of estimator withboth low complexity, good channel tracking ability and optimal placementof pilots. The criteria used to analyze the performance oftraining-based channel estimates can be classified into two areas:

1) information theoretic (mutual information and channel capacitybounds, cut-off rate)

2) signal processing (channel mean-square error (MSE), symbol MSE, biterror rate (BER))

In large MIMO systems, a key question of interest in channel estimatesis how much time should be spent in training, for a given number oftransmit and receive antennas, length of coherence time (T) and averagereceived SNR. The trade-off between the quality of channel estimate andinformation throughput plays an important role in selection of optimaltraining-based schemes. Hence, channel accesses employed for pilottransmission and for data transmission needs to be optimized for totalthroughput and fairness of the system.

In massive MIMO, the large number of antennas necessitates the use ofchannel estimates that have low computational complexity and highaccuracy. Where the two constraints work against each other, a goodtrade-off is required for high data rates and low channel estimationerrors. QLO modulation can provide solutions for both. The minimum meansquare error (MMSE) and minimum variance unbiased (MVU) channelestimates have high computational complexity for massive MIMO systemscompared to the low complexity schemes

TDD System

In TDD systems, BSs 302 and UTs 306 share the same frequency band fortransmission; hence, it is considered an efficient way to obtain CSI forfast changing channels. A distinguishing feature of TDD systems isreciprocity, where it is assumed that the forward channel is equal tothe transpose of the reverse channel. This eliminates the need forfeedback and allows for acquisition of CSI through reciprocity ofwireless medium with UL training signals. The use of a TDD scheme makesa massive MIMO system scalable in the number of service antennas to adesired extent, although the constraint of coherence interval needs tobe considered.

The communication is divided into two phases: the UL phase and DL phase.In the UL phase, UTs 306 transmit pilot signals to the BS 302, the BSuses these pilot signals for channel estimate processes and to form apre-coding matrices. The produced matrices are used to transmitpre-coded data to the UTs 306 located in each BS 302 cell in the DLphase. In a multi-cell scenario, non-orthogonal pilots acrossneighboring cells are utilized, as orthogonal pilots would need lengthof at least K×L symbols (K=total number of UTs in a cell and L=totalnumber of cell in the system) owing to frequency reuse factor of 1. Theuse of a K×L symbols training sequence is not feasible in practice formulti-cell as a result of short channel coherence times due to mobilityof UTs. This causes a phenomenon known as pilot contamination and it isas a major impairment in the performance of massive MIMO systems. Thisphenomenon introduces a finite SIR (signal to interference ratio) to thenetwork, which in turn, causes saturation effect i.e. the systemthroughput does not grow with the number of BS antennas.

FDD System

The FDD system is described to illustrate the cost of obtaining CSIthrough feedback in terms of system resources and bandwidth. In FDDsystems, the pre-coding in the DL and detection in the UL use differentfrequency bands. Therefore, the use of feedback is required in gettingthe CSI. In the process of obtaining the CSI in the DL, the BS 302 firsttransmits pilot symbols to all UTs 306, and the UTs feedback theestimated CSI (partial or complete) to the BS for the DL channels. Thefeedback resources used in FDD multi-user diversity system scales withthe number of antennas and will therefore grow large in a massive MIMOsystem with hundreds of antennas which leads to a loss of time-frequencyresources. As a result, the overhead in FDD becomes large compared toTDD systems where the overhead scales only with the number of users. Forfeedback cost and spectral efficiency, the data bandwidth decreases withthe number of UTs 306. This is a major challenge in the deployment ofmassive MIMO system using FDD mode and is also attracting a lot ofinterest because an FDD system is popular among network providers in theUSA.

Mitigating Pilot Contamination

The proposed methods for mitigating pilot channel contamination can beclassified into two categories, namely, a pilot-based estimationapproach and a subspace-based estimation approach. In the pilot-basedapproach, channels of UTs 306 are estimated using orthogonal pilotswithin the cell and non-orthogonal pilots across the cells. In thesubspace-based estimation approach, the channels of UTs 306 areestimated with or without limited pilots.

Mitigating Pilot Contamination-Pilot Estimation Approach

A time-shifted protocol for pilot transmission can reduce pilotcontamination in multi-user TDD systems. The transmission of pilotsignals in each cell is done by shifting the pilot locations in framesso that users in different cells transmit at non-overlapping times.Pilot contamination can be eliminated using the shift method as long aspilots do not overlap in time. The use of power allocation algorithms incombination with the time-shifted protocol can provide significantgains. Although the method looks promising, a major challenge inpractice is the control mechanism needed to dynamically synchronize thepilots across several cells so that they do not overlap. It is importantto note that due to the emergence of multi-tier heterogeneous cellularnetworks and dynamic placement of small cells, there will always beoverlap in time and frequency somewhere in the network.

Another process is a covariance aided channel estimation method thatexploits the covariance information of both desired and interfering userchannels. In the ideal case where the desired and the interferencecovariance span distinct subspaces, the pilot contamination effect tendsto vanish in the large antenna array case. As a result, users withmutually non-overlapping angle of arrival (AoA) hardly contaminate eachother. Therefore, one can perform a coordinated pilot assignment basedon assigning carefully selected groups of users to identical pilotsequences.

A similar approach can be used in a cognitive massive MIMO system.Although this method shows a significant reduction in inter-cellinterference and a corresponding increase in UL and DL SINRs, inpractice it is difficult to implement because it requires second orderstatistics of all the UL channels.

A spatial domain method like AoA or direction of arrival (DoA) can alsobe used where the channel coefficient of the strongest UL path is chosenas the DL beamformer. Based on similar assumptions made that the AoAs ofUTs are non-overlapping, an angular tunable predetermined scheme oroffline generated codebook can be used to match UL paths and then usedas a DL beam vector with the goal of avoiding leaked signals to UTs inadjacent cells. However, in this method the coherence interval needs tobe considered while searching for the optimal steering vector to be usedfor DL beamforming.

Therefore, one can use a pilot contamination elimination scheme whichrelies on two processing stages namely DL training and scheduled ULtraining. In the DL stage, UT supported by each BS estimates theirspecific DL frequency-domain channel transfer functions from the DLpilots of the BS. In the scheduled UL training stage, from each cell ata time, the UTs use the estimated DL frequency-domain channel transferfunctions to pre-distort their UL pilot symbols in which theuncontaminated DL channel transfer functions are ‘encapsulated’ in theUL pilot symbols for exploitation at the BS. Thereafter, the BS extractsall the DL-FDCHTFs of its UTs from the received UL signals byeliminating UL pilot signals of UTs from all other cells, hence,eliminating pilot contamination.

A major drawback of this scheme is the cost of overhead used intraining, which in reality can increase infinitely. As a result, apre-coding matrix at each BS is designed to minimize the sum of themean-square error of signals received at the UTs in the same cell andthe mean-square interference occurred at the UTs in other cells. Thistechnique offers significant performance gains and reduces theinter-cell and intra-cell interference compared to conventionalsingle-cell pre-coding method. However, the method assumes all the UTsare the same without differentiating them based on channels.

There is also a possibility of a pilot contamination pre-coding (PCP)method, which involves limited collaboration between BSs. In the PCPmethod, the first BS shares the slow-fading coefficient estimate withthe other BS or to a network hub, which computes the PCP pre-codingmatrices. The computed pre-coded matrices are forwarded to eachcorresponding BS for computation of the transmitted signal vectorsthrough its M antennas. This process is performed in the UL and DL. Theeffectiveness of this method lies in the accuracy of the sharedinformation from each BS and the computation of PCP by the network hub.One can also extend PCP method by an outer multi-cellular pre-codingcalled large-scale fading pre-coding (LSFP) and large scale fadingdecoding (LSFD) with a finite number of BS antennas. These methods aredesigned to maximize the minimum rate with individual BS powerconstraints which delivers significant improvement on the 5% outage ratecompared to existing methods.

Mitigating Pilot Contamination-Subspace Based Estimation Approach

Subspace-based channel estimation techniques are a promising approachfor increased spectral efficiency because it requires no or a verylimited number of pilot symbols for operation. In this approach, signalproperties, such as finite alphabet structure, fixed symbol rate,constant modulus, independence, and higher order statistical properties,can be used for channel estimation. This approach can be extended tochannel estimation in multi-cell TDD systems with the focus ofeliminating pilot contamination. CSI is obtained by applying a subspaceestimation technique using eigenvalue decomposition (EVD) on thecovariance matrix of the received samples, but up to a scalar ambiguity.

To overcome this ambiguity, short training orthogonal pilots areintroduced in all the cells. The EVD-based method is prone to error dueto the assumption that channel vectors between the users and the BSbecome pair-wisely orthogonal when the number of BS antennas M tendstowards infinity. However, in practice M is large but finite. To reducethese errors, the EVD algorithm is combined with the iterativeleast-square with projection algorithms. The EVD method is not affectedby pilot contamination and performs better than conventional pilot-basedtechniques but its accuracy depends on large number of BS antennas andincreased sampling data within the coherence time.

There are other blind methods for channel estimation in a cellularsystems with power control and power controlled hand-off. The main ideais to find the singular value decomposition of the received signalmatrix and to determine which system parameters in the subspace of thesignal of interest can be identified blindly using approximate analysisfrom random matrix theory. In most cases it is sufficient to know thesubspace which the channel vectors of interest span, in order to acquireaccurate channel estimates for the projected channel. However, thelimitation of this approach in practice is that the assumption that alldesired channels are stronger than all interfering channels does notalways hold. To overcome this limitation, a maximum a-posteriori (MAP)criterion for subspace channel estimation can be used. The MAP methodcan be more robust and offers better performance than the blind methodbut with increased complexity.

A diagonal jacket-based estimation method with iterative least-squareprojection can be used for fast channel estimation and reduction ofpilot contamination problems. The BS correlates the received pilottransmissions which are corrupted by pilot transmissions from othercells to produce its channel estimates. As the geometric attenuationfrom neighboring cells increases, the system performance of aconventional pilot based system degrades due to pilot contaminationwhereas the diagonal jacket matrix is not affected.

Mitigating Pilot Contamination—a New Orthogonal Set

Referring now to FIG. 4, there is illustrated an approach of mitigatingpilot contamination using a new orthogonal basis set by combiningmultilevel overlay 402 with a massive MIMO system 404, as described inU.S. patent Ser. No. 15/216,474, entitled SYSTEM AND METHOD FORCOMBINING MIMO AND MODE-DIVISION MULTIPLEXING, filed on Jul. 21, 2016,which is incorporated herein by reference, to provide improved pilotchannel contamination 406. In this method, the transmission of pilotsignals in each cell is done by a set of pilots that minimize thetime-frequency product or resources. These signals do not have anycorrelations with one another neither in time domain or frequencydomain. Pilot contamination can be eliminated using these pilot signalsas all versions of the signals are mutually orthogonal to any other fromwithin the cell or outside of the cell.

As illustrated in FIG. 5, a pilot signal 502 is transmitted between atransmitter 504 (normally as user terminal) to a receiver 506 (normallyas base station. The pilot signal includes an impulse signal that isreceived, detected and processed at the receiver 506. Using theinformation received from the pilot impulse signal, the channel 508between the transmitter 504 and receiver 506 may be processed todetermine channel state information at the receiver/base station 506 andremove noise, fading and other channel impairment issues from thechannel 508. When multiple pilots 502 are transmitted to a receiver 506rather than a single receiver, pilot signals 502 will interfere witheach other causing pilot channel contamination.

This process is generally described with respect to the flowchart ofFIG. 6. The pilot impulse signal is transmitted at 602 over thetransmission channel. The impulse response is detected at step 604 andprocessed to determine the channel state information and impulseresponse over the transmission channel. Effects of channel impairmentssuch as noise and fading may be countered by multiplying signalstransmitted over the transmission channel by the inverse of the impulseresponse at step 606 in order to correct for the various channelimpairments that may be up on the transmission channel. In this way thechannel impairments are counteracted and improved signal quality andreception may be provided over the transmission channel.

Cross talk and multipath interference can be corrected using RFMultiple-Input-Multiple-Output (MIMO). Most of the channel impairmentscan be detected using a control or pilot channel and be corrected usingalgorithmic techniques (closed loop control system). Interferencebetween the pilot channel signals can be overcome by modulating thepilot signals according to the multiple level overlay/quantum leveloverlay (MLO/QLO) techniques described herein below. The modulationusing the MLO/QLO techniques minimize the time-bandwidth product andprevent interference between the different pilot signals.

Within the notational two-dimensional space, minimization of the timebandwidth product, i.e., the area occupied by a signal in that space,enables denser packing, and thus, the use of more signals, with higherresulting information-carrying capacity and less cross channelinterference, between allocated channel. Given the frequency channeldelta (Δf), a given signal transmitted through it in minimum time Δtwill have an envelope described by certain time-bandwidth minimizingsignals. The time-bandwidth products for these signals take the form;ΔtΔf=½(2n+1)where n is an integer ranging from 0 to infinity, denoting the order ofthe signal.

These signals form an orthogonal set of infinite elements, where eachhas a finite amount of energy. They are finite in both the time domainand the frequency domain, and can be detected from a mix of othersignals and noise through correlation, for example, by match filtering.Unlike other wavelets, these orthogonal signals have similar time andfrequency forms. These types of orthogonal signals that reduce the timebandwidth product and thereby increase the spectral efficiency of thechannel. They also prevent interference between pilot channels.

Hermite-Gaussian polynomials are one example of a classical orthogonalpolynomial sequence, which are the Eigenstates of a quantum harmonicoscillator. Signals based on Hermite-Gaussian polynomials possess theminimal time-bandwidth product property described below, and may be usedfor embodiments of MLO systems. However, it should be understood thatother signals may also be used, for example orthogonal polynomials suchas Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials,Chebyshev polynomials, Laguerre-Gaussian polynomials, Hermite-Gaussianpolynomials and Ince-Gaussian polynomials. Q-functions are another classof functions that can be employed as a basis for MLO signals.

In addition to the time bandwidth minimization described above, theplurality of data streams can be processed to provide minimization ofthe Space-Momentum products in spatial modulation. In this case:ΔxΔp=½

Referring now to FIG. 7, there is illustrated an example of a systemutilizing MIMO transmissions between a base station 702 and a pluralityof user devices 704. Each of the base station 702 and the user devices704 include a transceiver 706 enabling transmission of RF or opticalwireless communications between the base station 702 and the pluralityof user devices 704. The base station 702 further has associated therewith a MIMO array 708 including a plurality of transmitting andreceiving antennas. Each of the transceivers 706 include the necessarycircuitry and control components for generating multilevel overlaymodulation within both signal transmissions and more particularly foruse with pilot channel transmissions from the user devices 704 to/fromthe base station 702. These MLO components 710 provide for theapplication of multilevel overlay modulation to the pilot channeltransmissions and other channel transmissions as described herein. Byapplying the MLO modulation, each of the pilot channel transmissions 712between the user devices 704 and the base station 702 will not causepilot channel interference between the various pilot channeltransmissions 712 and enable a greater number of pilot channeltransmissions using the MIMO array 708 for reception/transmissionwithout increase in the pilot channel contamination between the variouspilot channels 712.

Referring now to FIG. 8, there is illustrated a flow diagram generallydescribing the process for generating pilot channel 712 communicationsfrom user devices 704 to a base station 702 implementing a MIMO array708. Initially, at step 802, the user devices 704 generate pilot signalsfor transmission to the base station 702. The generated pilot signalsare modulated at step 804 using MLO/QLO modulation techniques prior totheir transmission to the base station. Each of the MLO/QLO modulatedpilot signals are transmitted from their respective user devices 704 tothe base station 702 at step 806. The received pilot signals aredemodulated at step 808 to remove the MLO/QLO modulation. Thedemodulated pilot signals are used at step 810 for generating channelstate information for the pilot channels to the base station 702. Thebase station 702 may then utilize generated channel state informationfor establishing channels between the base station 702 in the userdevices 704. Communications may then be carried out over the establishcommunication channels at step 814. The transmissions of the pilotsignals may also occur from the base station 702 to the user device 704in a similar fashion.

This system and method for improvement of pilot channel contaminationinvolves the implementation of two major structures. These include amassive MIMO signal transmission system involving a base station 702including an array of antennas for carrying out transmissions tomultiple user devices 704 and a multi-level overlay/quantum leveloverlay (MLO/QLO) modulation system for modulating pilot channel signalsthat are transmitted between the user devices and the base station. Eachof these are discussed more fully herein below.

Massive MIMO System

As described above, a further manner for limiting contamination of pilotchannels is the combination of multiple-input multiple-output(MIMO)-based spatial multiplexing and multiple level overlay (MLO)modulation. Such a combined MIMO+MLO can enhance the performance ofpilot channels in free-space Point-to-Point communications systems. Thiscan be done at both RF as well as optical frequencies. Inter-channelcrosstalk effects on the pilot channels can be minimized by the inherentorthogonality of the MLO modulation and by the use of MIMO signalprocessing.

When multiple input/multiple output (MIMO) systems were described in themid-to-late 1990s by Dr. G. Foschini and Dr. A. Paulraj, the astonishingbandwidth efficiency of such techniques seemed to be in violation of theShannon limit. But, there is no violation of the Shannon limit becausethe diversity and signal processing employed with MIMO transforms apoint-to-point single channel into multiple parallel or matrix channels,hence in effect multiplying the capacity. MIMO offers higher data ratesas well as spectral efficiency. This is more particularly So illustratedin FIG. 9 wherein a single transmitting antenna 902 transmits to asingle receiving antenna 904 using a total power signal Ptotal. The MIMOsystem illustrated in FIG. 10 provides the same total power signalPtotal to a multi-input transmitter consisting of a plurality ofantennas 1002. The receiver includes a plurality of antennas 1004 forreceiving the transmitted signal. Many standards have alreadyincorporated MIMO. ITU uses MIMO in the High Speed Downlink PacketAccess (HSPDA), part of the UMTS standard. MIMO is also part of the802.11n standard used by wireless routers as well as 802.16 for MobileWiMax, LTE, LTE Advanced and future 5G standards.

A traditional communications link, which is called asingle-in-single-out (SISO) channel as shown in FIG. 9, has onetransmitter 902 and one receiver 904. But instead of a singletransmitter and a single receiver several transmitters 1002 andreceivers 1004 may be used as shown in FIG. 10. The SISO channel thusbecomes a multiple-in-multiple-out, or a MIMO channel; i.e. a channelthat has multiple transmitters and multiple receivers.

The capacity of a SISO link is a function simply of the channel SNR asgiven by the Equation: C=log₂ (1+SNR). This capacity relationship was ofcourse established by Shannon and is also called theinformation-theoretic capacity. The SNR in this equation is defined asthe total power divided by the noise power. The capacity is increasingas a log function of the SNR, which is a slow increase. Clearlyincreasing the capacity by any significant factor takes an enormousamount of power in a SISO channel. It is possible to increase thecapacity instead by a linear function of power with MIMO.

With MIMO, there is a different paradigm of channel capacity. If sixantennas are added on both transmit and receive side, the same capacitycan be achieved as using 100 times more power than in the SISO case. Thetransmitter and receiver are more complex but have no increase in powerat all. The same performance is achieved in the MIMO system as isachieved by increasing the power 100 times in a SISO system.

In FIG. 11, the comparison of SISO and MIMO systems using the samepower. MIMO capacity 1102 increases linearly with the number ofantennas, where SISO/SIMO/MISO systems 1104 all increase onlylogarithmically.

At conceptual level, MIMO enhances the dimensions of communication.However, MIMO is not Multiple Access. It is not like FDMA because all“channels” use the same frequency, and it is not TDMA because allchannels operate simultaneously. There is no way to separate thechannels in MIMO by code, as is done in CDMA and there are no steerablebeams or smart antennas as in SDMA. MIMO exploits an entirely differentdimension.

A MIMO system provides not one channel but multiple channels, NR×NT,where NT is the number of antennas on the transmit side and NR, on thereceive side. Somewhat like the idea of OFDM, the signal travels overmultiple paths and is recombined in a smart way to obtain these gains.

In FIG. 12 there is illustrated a comparison of a SISO channel 1202 with2 MIMO channels 1204, 1206, (2×2) and (4×4). At SNR of 10 dB, a 2×2 MIMOsystem 1204 offers 5.5 b/s/Hz and whereas a 4×4 MIMO link offers over 10b/s/Hz. This is an amazing increase in capacity without any increase intransmit power caused only by increasing the number of transceivers. Notonly that, this superb performance comes in when there are channelimpairments, those that have fading and Doppler.

Extending the single link (SISO) paradigm, it is clear that to increasecapacity, a link can be replicated N times. By using N links, thecapacity is increased by a factor of N. But this scheme also uses Ntimes the power. Since links are often power-limited, the idea of N linkto get N times capacity is not much of a trick. Can the number of linksbe increased but not require extra power? How about if two antennas areused but each gets only half the power? This is what is done in MIMO,more transmit antennas but the total power is not increased. Thequestion is how does this result in increased capacity?

The information-theoretic capacity increase under a MIMO system is quitelarge and easily justifies the increase in complexity as illustrated inFIG. 13. First and second transmitters 1302 transmit to a pair ofreceivers 1304. Each of the transmitters 1302 has a transmission link1306 to an antenna of a receiver 1304. Transmitter TX#1 transmits onlink h11 and h21. Transmitter TX#2 transmits on links h12 and h22. Thisprovides a matrix of transmission capacities according to the matrix:

$H = \begin{bmatrix}h_{11} & h_{12} \\h_{21} & h_{22}\end{bmatrix}$And a total transmission capacity of according to the equation:

$C = {\max\limits_{{{tr}{(R_{xx})}} = P_{T}}{{\log\det}\left\{ {I_{N} + {\frac{I}{\sigma_{n}^{2}}{HR}_{xx}H^{H}}} \right\}}}$

In simple language, MIMO is any link that has multiple transmit andreceive antennas. The transmit antennas are co-located, at a little lessthan half a wavelength apart or more. This figure of the antennaseparation is determined by mutual correlation function of the antennasusing Jakes Model. The receive antennas 1304 are also part of one unit.Just as in SISO links, the communication is assumed to be between onesender and one receiver. MIMO is also used in a multi-user scenario,similar to the way OFDM can be used for one or multiple users. Theinput/output relationship of a SISO channel is defined as:r=hs+nwhere r is the received signal, s is the sent signal and h, the impulseresponse of the channel is n, the noise. The term h, the impulseresponse of the channel, can be a gain or a loss, it can be phase shiftor it can be time delay, or all of these together. The quantity h can beconsidered an enhancing or distorting agent for the signal SNR.

Referring now to FIG. 14 there are illustrated various types of multipleinput and multiple output transmission systems. System 1402 illustratesa single input single output SISO system. System 1404 illustrates asingle input multiple output receiver SIMO system. System 1406illustrates a multiple input single output MISO system. Finally, amultiple input multiple output MIMO system is illustrated at 1410. Thechannels of the MIMO system 1410 can be thought of as a matrix channel.

Using the same model a SISO, MIMO channel can now be described as:R=HS+N

In this formulation, both transmit and receive signals are vectors. Thechannel impulse response h, is now a matrix, H. This channel matrix H iscalled Channel Information. The channel matrix H can be created using apilot signal over a pilot channel in the manner described herein above.The signals on the pilot channel may be sent in a number of differentforms such as HG beams, LG beams or other orthogonal beams of any order.

Dimensionality of Gains in MIMO

The MIMO design of a communications link can be classified in these twoways.

-   -   MIMO using diversity techniques    -   MIMO using spatial-multiplexing techniques        Both of these techniques are used together in MIMO systems. With        first form, Diversity technique, same data is transmitted on        multiple transmit antennas and hence this increases the        diversity of the system.

Diversity means that the same data has traveled through diverse paths toget to the receiver. Diversity increases the reliability ofcommunications. If one path is weak, then a copy of the data received onanother path may be just fine.

FIG. 15 illustrates a source 1502 with data sequence 101 to be sent overa MIMO system with three transmitters. In the diversity form 1504 ofMIMO, same data, 101 is sent over three different transmitters. If eachpath is subject to different fading, the likelihood is high that one ofthese paths will lead to successful reception. This is what is meant bydiversity or diversity systems. This system has a diversity gain of 3.

The second form uses spatial-multiplexing techniques. In a diversitysystem 1504, the same data is sent over each path. In aspatial-multiplexing system 1506, the data 1,0,1 is multiplexed on thethree channels. Each channel carries different data, similar to the ideaof an OFDM signal. Clearly, by multiplexing the data, the datathroughput or the capacity of the channel is increased, but thediversity gain is lost. The multiplexing has tripled the data rate, sothe multiplexing gain is 3 but diversity gain is now 1. Whereas in adiversity system 1504 the gain comes in form of increased reliability,in a spatial-multiplexing system 1506, the gain comes in the form of anincreased data rate.

Characterizing a MIMO Channel

When a channel uses multiple receive antennas, N_(R), and multipletransmit antennas, N_(T), the system is called a multiple-input,multiple output (MIMO) system.

When N_(T)=N_(R)=1, a SISO system.

When N_(T)>1 and N_(R)=1, called a MISO system,

When N_(T)=1 and N_(R)>1, called a SIMO system.

When N_(T)>1 and N_(R)>1, is a MIMO system.

In a typical SISO channel, the data is transmitted and reception isassumed. As long as the SNR is not changing dramatically, no questionsare asked regarding any information about the channel on a bit by bitbasis. This is referred to as a stable channel. Channel knowledge of aSISO channel is characterized only by its steady-state SNR.

What is meant by channel knowledge for a MIMO channel? Assume a linkwith two transmitters and two receivers on each side. The same symbol istransmitted from each antenna at the same frequency, which is receivedby two receivers. There are four possible paths as shown in FIG. 13.Each path from a transmitter to a receiver has some loss/gain associatedwith it and a channel can be characterized by this loss. A path mayactually be sum of many multipath components but it is characterizedonly by the start and the end points. Since all four channels arecarrying the same symbol, this provides diversity by making up for aweak channel, if any. In FIG. 16 there is illustrated how each channelmay be fading from one moment to the next. At time 32, for example, thefade in channel h₂₁ is much higher than the other three channels.

As the number of antennas and hence the number of paths increase in aMIMO system, there is an associated increase in diversity. Thereforewith the increasing numbers of transmitters, all fades can probably becompensated for. With increasing diversity, the fading channel starts tolook like a Gaussian channel, which is a welcome outcome.

The relationship between the received signal in a MIMO system and thetransmitted signal can be represented in a matrix form with the H matrixrepresenting the low-pass channel response h_(ij), which is the channelresponse from the j_(th) antenna to the i_(th) receiver. The matrix H ofsize (N_(R), N_(T)) has N_(R) rows, representing N_(R) received signals,each of which is composed of N_(T) components from N_(T) transmitters.Each column of the H matrix represents the components arriving from onetransmitter to N_(R) receivers.

The H matrix is called the channel information. Each of the matrixentries is a distortion coefficient acting on the transmitted signalamplitude and phase in time-domain. To develop the channel information,a symbol is sent from the first antenna, and a response is noted by allthree receivers. Then the other two antennas do the same thing and a newcolumn is developed by the three new responses.

The H matrix is developed by the receiver. The transmitter typicallydoes not have any idea what the channel looks like and is transmittingblindly. If the receiver then turns around and transmits this matrixback to the transmitter, then the transmitter would be able to see howthe signals are faring and might want to make adjustments in the powersallocated to its antennas. Perhaps a smart computer at the transmitterwill decide to not transmit on one antenna, if the received signals areso much smaller (in amplitude) than the other two antennas. Maybe thepower should be split between antenna 2 and 3 and turn off antenna 1until the channel improves.

Modeling a MIMO Channel

Starting with a general channel which has both multipath and Doppler(the conditions facing a mobile in case of a cell phone system), thechannel matrix H for this channel takes this form.

${H\left( {\tau,t} \right)} = \begin{bmatrix}{h_{11}\left( {\tau,t} \right)} & {h_{11}\left( {\tau,t} \right)} & \ldots & {h_{1N_{T}}\left( {\tau,t} \right)} \\{h_{21}\left( {\tau,t} \right)} & {h_{22}\left( {\tau,t} \right)} & \ldots & {h_{2N_{T}}\left( {\tau,t} \right)} \\\vdots & \vdots & \ddots & \vdots \\{h_{N_{R},1}\left( {\tau,t} \right)} & {h_{N_{R},2}\left( {\tau,t} \right)} & \ldots & {h_{N_{R},N_{T}}\left( {\tau,t} \right)}\end{bmatrix}$

Each path coefficient is a function of not only time t because thetransmitter is moving but also a time delay relative to other paths. Thevariable τ indicates relative delays between each component caused byfrequency shifts. The time variable t represents the time-varying natureof the channel such as one that has Doppler or other time variations.

If the transmitted signal is s_(i)(t), and the received signal isr_(i)(t), the input-output relationship of a general MIMO channel isdefined as:

${{r_{i}(t)} = {{\sum\limits_{j = 1}^{N_{T}}{\int_{- \infty}^{\infty}{{h_{ij}\left( {\tau,t} \right)}{S_{j}\left( {t - \tau} \right)}{dt}}}} = {{\sum\limits_{j = 1}^{N_{T}}{{h_{ij}\left( {\tau,t} \right)}*{S_{j}(\tau)}\mspace{14mu} i}} = 1}}},{2\mspace{14mu}\ldots\mspace{14mu} N_{R}}$

The channel equation for the received signal r_(i)(t) is expressed as aconvolution of the channel matrix H and the transmitted signals becauseof the delay variable τ. This relationship can be defined in matrix formas:r(t)=H(τ,t)*s(t)

If the channel is assumed to be flat (non-frequency selective), but istime-varying, i.e. has Doppler, the relationship is written without theconvolution as:r(t)=H(t)s(t)

In this case, the H matrix changes randomly with time. If the timevariations are very slow (non-moving receiver and transmitter) such thatduring a block of transmission longer than the several symbols, thechannel can be assumed to be non-varying, or static. A fixed realizationof the H matrix for a fixed Point-to-Point scenario can be written as:r(t)=H(t)s(t)The individual entries can be either scalar or complex.

For analysis purposes, important assumptions can be made about the Hmatrix. We can assume that it is fixed for a period of one or moresymbols and then changes randomly. This is a fast change and causes theSNR of the received signal to change very rapidly. Or we can assume thatit is fixed for a block of time, such as over a full code sequence,which makes decoding easier because the decoder does not have to dealwith a variable SNR over a block. Or we can assume that the channel issemi-static such as in a TDMA system, and its behavior is static over aburst or more. Each version of the H matrix seen is a realization. Howfast these realizations change depends on the channel type.

$H = \begin{bmatrix}h_{11} & h_{12} & \ldots & h_{1N_{T}} \\h_{21} & h_{22} & \ldots & h_{2N_{T}} \\\vdots & \vdots & \ddots & \vdots \\h_{N_{R},1} & h_{N_{R},2} & \ldots & h_{N_{R},1}\end{bmatrix}$

For a fixed random realization of the H matrix, the input-outputrelationship can be written without the convolution as:r(t)=H s(t)

In this channel model, the H matrix is assumed to be fixed. An exampleof this type of situation where the H matrix may remain fixed for a longperiod would be a Point-to-Point system where we have fixed transmitterand receiver. In most cases, the channel can be considered to be static.This allows us to treat the channel as deterministic over that periodand amenable to analysis. In a point-to-point system, the channel issemi-static and it behavior is static over a burst or more. Each versionof the H matrix is a realization.

The power received at all receive antennas is equal to the sum of thetotal transmit power, assuming channel offers no gain or loss. Eachentry h_(ij) comprises an amplitude and phase term. Squaring the entryh_(ij) give the power for that path. There are N_(T) paths to eachreceiver, so the sum of j terms, provides the total transmit power. Eachreceiver receives the total transmit power. For this relation, thetransmit power of each transmitter is assumed to be 1.

${\sum\limits_{j = 1}^{N_{T}}\left( h_{ij} \right)^{2}} = {{N_{T}(1)} = N_{T}}$

The H matrix is a very important construct in understanding MIMOcapacity and performance. How a MIMO system performs depends on thecondition of the channel matrix H and its properties. The H matrix canbe thought of as a set of simultaneous equations. Each equationrepresents a received signal which is a composite of unique set ofchannel coefficients applied to the transmitted signal.r ₁ =h ₁₁ s+h ₁₂ s . . . +h _(1N) ₂ s

If the number of transmitters is equal to the number of receivers, thereexists a unique solution to these equations. If the number of equationsis larger than the number of unknowns (i.e. N_(R)>N_(T)), the solutioncan be found using a zero-forcing algorithm. When N_(T)=N_(R), (thenumber of transmitters and receivers are the same), the solution can befound by (ignoring noise) inverting the H matrix as in:{circumflex over (s)}(t)=H ⁻¹ r(t)

The system performs best when the H matrix is full rank, with eachrow/column meeting conditions of independence. What this means is thatbest performance is achieved only when each path is fully independent ofall others. This can happen only in an environment that offers richscattering, fading and multipath, which seems like a counter-intuitivestatement. Looking at the equation above, the only way to extract thetransmitted information is when the H matrix is invertible. And the onlyway it is invertible is if all its rows and columns are uncorrelated.And the only way this can occur is if the scattering, fading and allother effects cause the channels to be completely uncorrelated.

Diversity Domains and MIMO Systems

In order to provide a fixed quality of service, a large amount oftransmit power is required in a Rayleigh or Rican fading environment toassure that no matter what the fade level, adequate power is stillavailable to decode the signal. Diversity techniques that mitigatemultipath fading, both slow and fast are called Micro-diversity, whereasthose resulting from path loss, from shadowing due to buildings etc. arean order of magnitude slower than multipath, are called Macro-diversitytechniques. MIMO design issues are limited only to micro-diversity.Macro-diversity is usually handled by providing overlapping base stationcoverage and handover algorithms and is a separate independentoperational issue.

In time domain, repeating a symbol N times is the simplest example ofincreasing diversity. Interleaving is another example of time diversitywhere symbols are artificially separated in time so as to createtime-separated and hence independent fading channels for adjacentsymbols. Error correction coding also accomplishes time-domain diversityby spreading the symbols in time. Such time domain diversity methods aretermed here as temporal diversity.

Frequency diversity can be provided by spreading the data overfrequency, such as is done by spread spectrum systems. In OFDM frequencydiversity is provided by sending each symbol over a different frequency.In all such frequency diversity systems, the frequency separation mustbe greater than the coherence bandwidth of the channel in order toassure independence.

The type of diversity exploited in MIMO is called Spatial diversity. Thereceive side diversity, is the use of more than one receive antenna. SNRgain is realized from the multiple copies received (because the SNR isadditive). Various types of linear combining techniques can take thereceived signals and use special combining techniques such are MaximalRatio Combining, Threshold Combing etc. The SNR increase is possible viacombining results in a power gain. The SNR gain is called the arraygain.

Transmit side diversity similarly means having multiple transmitantennas on the transmit side which create multiple paths and potentialfor angular diversity. Angular diversity can be understood asbeam-forming. If the transmitter has information about the channel, asto where the fading is and which paths (hence direction) is best, thenit can concentrate its power in a particular direction. This is anadditional form of gain possible with MIMO.

Another form of diversity is Polarization diversity such as used insatellite communications, where independent signals are transmitted oneach polarization (horizontal vs. vertical). The channels, although atthe same frequency, contain independent data on the two polarized henceorthogonal paths. This is also a form of MIMO where the two independentchannels create data rate enhancement instead of diversity. So satellitecommunications is a form of a (2, 2) MIMO link.

Related to MIMO

There are some items that need be explored as they relate to MIMO butare usually not part of it. First are the smart antennas used in set-topboxes. Smart antennas are a way to enhance the receive gain of a SISOchannel but are different in concept than MIMO. Smart antennas usephased-arrays to track the signal. They are capable of determining thedirection of arrival of the signal and use special algorithms such asMUSIC and MATRIX to calculate weights for its phased arrays. They areperforming receive side processing only, using linear or non-linearcombining.

Rake receivers are a similar idea, used for multipath channels. They area SISO channel application designed to enhance the received SNR byprocessing the received signal along several “fingers” or correlatorspointed at particular multipath. This can often enhance the receivedsignal SNR and improve decoding. In MIMO systems Rake receivers are notnecessary because MIMO can actually simplify receiver signal processing.

Beamforming is used in MIMO but is not the whole picture of MIMO. It isa method of creating a custom radiation pattern based on channelknowledge that provides antenna gains in a specific direction. Beamforming can be used in MIMO to provide further gains when thetransmitter has information about the channel and receiver locations.

Importance of Channel State Information

Dealing with Channel matrix H is at the heart of how MIMO works. Ingeneral, the receiver is assumed to be able to get the channelinformation easily and continuously. It is not equally feasible for thetransmitter to obtain a fresh version of the channel state information,because the information has gotten impaired on the way back. However, aslong as the transit delay is less than channel coherence time, theinformation sent back by the receiver to the transmitter retains itsfreshness and usefulness to the transmitter in managing its power. Atthe receiver, we refer to channel information as Channel State (or side)Information at the Receiver, CSIR. Similarly when channel information isavailable at the transmitter, it is called CSIT. CSI, the channel matrixcan be assumed to be known instantaneously at the receiver or thetransmitter or both. Although in short term the channel can have anon-zero mean, it is assumed to be zero-mean and uncorrelated on allpaths. When the paths are correlated, then clearly, less information isavailable to exploit. But the channel can still be made to work.

Channel information can be extracted by monitoring the received gains ofa known sequence over, for example, a pilot channel. In Time DivisionDuplex (TDD) communications where both transmitter and the receiver areon the same frequency, the channel condition is readily available to thetransmitter. In Frequency Division Duplex (FDD) communications, sincethe forward and reverse links are at different frequencies, thisrequires a special feedback link from the receiver to the transmitter.In fact receive diversity alone is very effective but it places greaterburden on the smaller receivers, requiring larger weight, size andcomplex signal processing hence increasing cost.

Transmit diversity is easier to implement in a cellular system from asystem point of view because the base station towers in a cell systemare not limited by power or weight. In addition to adding more transmitantennas on the base station towers, space-time coding is also used bythe transmitters. This makes the signal processing required at thereceiver simpler.

MIMO Gains

The goal is to transmit and receive data over several independentlyfading channels such that the composite performance mitigates deep fadeson any of the channels. To see how MIMO enhances performance in a fadingor multipath channel, the BER for a BPSK signal is examined as afunction of the receive SNR.P _(e)≈

(√{square root over (2∥h∥ ² SNR)})The quantity (h²×SNR) is the instantaneous SNR.

Now assume that there are L possible paths, where L=N_(R)×N_(T), withN_(T)=number of transmitter and N_(R)=number of receive antennas. Sincethere are several paths, the average BER can be expressed as a functionof the average channel gain over all these paths. This quantity is theaverage gain over all channels, L.

${h}^{2} = {{{Avg}\left\lbrack {h_{1}}^{2} \right\rbrack} = {\sum\limits_{I = 1}^{L}\;{h_{1}}^{2}}}$

The average SNR can be rewritten as a product of two terms.

${{h}^{2}{SNR}} = {{\underset{\_}{L \times {SNR}} \cdot \frac{1}{L}}{h}^{2}}$

The first part L×SNR is a linear increase in SNR due to the L paths.This term is called by various names, including power gain, rate gain orarray gain. This term can also include beamforming gain. Henceincreasing the number of antennas increases the array gain directly bythe factor L. The second term is called diversity gain. This is theaverage gain over L different paths.

It seems intuitive that if one of the paths exhibits deep fading then,when averaged over a number of independent paths, the deep fades can beaveraged out. (We use the term channel to mean the composite of allpaths.) Hence on the average we would experience a diversity gain aslong as the path gains across the channels are not correlated. If thegains are correlated, such as if all paths are mostly line-of-sight, wewould obtain only an array gain and very little diversity gain. This isintuitive because a diversity gain can come only if the paths arediverse, or in other words uncorrelated.

MIMO Advantages

Operating in Fading Channels

The most challenging issue in communications signal design is how tomitigate the effects of fading channels on the signal BER. A fadingchannel is one where channel gain is changing dramatically, even at highSNR, and as such it results in poor BER performance as compared to anAWGN channel. For communications in a fading channel, a way to convertthe highly variable fading channel to a stable AWGN-like channel isneeded.

Multipath fading is a phenomenon that occurs due to reflectors andscatters in the transmission path. The measure of multipath is DelaySpread, which is the RMS time delay as a function of the power of themultipath. This delay is converted to a Coherence Bandwidth (CB), ametric of multipath. A time delay is equivalent to a frequency shift inthe frequency domain. So any distortion that delays a signal, changesits frequency. Thus, delay spread>bandwidth distortion.

Whether a signal is going through a flat or a frequency-selective fadingat any particular time is a function of coherence bandwidth of thechannel as compared with its bandwidth as shown in Table I below. If theCoherence bandwidth of the channel is larger than the signal bandwidth,then we have a flat or a non-frequency selective channel. What coherencemeans is that all the frequencies in the signal respond similarly or aresubject to the same amplitude distortion. This means that fading doesnot affect frequencies differentially, which is a good thing.Differential distortion is hard to deal with. So of all types of fading,flat fading is the least problematic.

The next source of distortion is Doppler. Doppler results in differentdistortions to the frequency band of the signal. The measure of Dopplerspread is called Coherence Time (CT) (no relationship to CoherenceBandwidth from the flat-fading case). The comparison of the CT with thesymbol time determines the speed of fading. So if the coherence time isvery small, compared to the symbol time, that's not good.

The idea of Coherence Time and Coherence Bandwidth is often confused.Flatness refers to frequency response and not to time. So CoherenceBandwidth determines whether a channel is considered flat or not.

Coherence Time, on the other hand has to do with changes over time,which is related to motion. Coherence Time is the duration during whicha channel appears to be unchanging. One can think about Coherence Timewhen Doppler or motion is present. When Coherence Time is longer thansymbol time, then a slow fading channel is provided and when symbol timeis longer than Coherence time, a fast fading channel is provided. Soslowness and fastness mean time based fading.

TABLE I MIMO channel Types and their Measures Channel Channel SpreadSelectivity Type Measure Delay Spread Frequency Non-selective CoherenceBandwidth > Signal Bandwidth Frequency Selective Signal Bandwidth >Coherence Bandwidth Doppler Spread Time Slow-fading Coherence Time >Symbol Time Fast-fast fading Symbol Time > Coherence Time Angle SpreadBeam pattern — Coherence Distance

In addition to these fast channel effects, there are mean path losses aswell as rain losses, which are considered order-of-magnitude slowereffects and are managed operationally and so will not be part of theadvantages of MIMO.

How MIMO Creates Performance Gains in a Fading Channel

Shannon defines capacity of a channel as a function of its SNR.Underlying this is the assumption that the SNR is invariant. For such asystem, Shannon capacity is called its ergodic capacity. Since SNR isrelated to BER, the capacity of a channel is directly related to howfast the BER declines with SNR. The BER needs to decrease quickly withincreasing power.

The Rayleigh channel BER when compared to an AWGN channel for the sameSNR is considerably bigger and hence the capacity of a Rayleigh channelwhich is the converse of its BER, is much lower. Using the BER of a BPSKsignal as a benchmark, one can examine the shortfall of a Rayleighchannel and see how MIMO can help to mitigate this loss.

For a BPSK signal, the BER in an AWGN channel is given by setting∥h²∥=1.P _(e)≈

(√{square root over (2SNR)})The BER of the same channel in a Rayleigh channel is given by

$P_{e} = {{\frac{1}{2}\left( {1 - \sqrt{\frac{SNR}{1 + {SNR}}}} \right)} \approx \frac{1}{2\;{SNR}}}$

FIG. 17 shows the BER of an AWGN and a Rayleigh channel as a function ofthe SNR. The AWGN BER 1702 varies by the inverse of the square of theSNR, SNR⁻² and declines much faster than the Rayleigh channel 1704 whichdeclines instead by SNR⁻¹. Hence an increase in SNR helps the Rayleighchannel 1704 much less than it does an AWGN channel 1702. The Rayleighchannel 1704 improves much more slowly as more power is added.

FIG. 17 also illustrates that for a BER of 10⁻³, an additional 17 dB ofpower is required in a Rayleigh channel 1704. This is a very largedifferential, nearly 50 times larger than AWGN 1702. One way to bringthe Rayleigh curve 1704 closer to the AWGN curve 1702 (which forms alimit of performance) is to add more antennas on the receive or thetransmit-side hence making SISO into a MIMO system.

Starting with just one antenna, the number of receive antennas isincreased to N_(R), while keeping one transmitter, making it a SIMOsystem. By looking at the asymptotic BER at large SNR (large SNR has noformal definition, anything over 15 dB can be considered large), adetermination can be made of the gain caused by adding just one moreantenna to the N antennas. The ratio of the BER to the BER due to onemore antenna may then be determined.

$\frac{{BER}\left( {N,{SNR}} \right)}{{BER}\left( {{N + 1},{SNR}} \right)} = {{SNR}\left( {1 + \frac{1}{{2\; N} + 1}} \right)}$

The gain from adding one more antenna is equal to SNR multiplied by adelta increase in SNR. The delta increase diminishes as more and moreantennas are added. The largest gain is seen when going from a singleantenna to two antennas, (1.5 for going from 1 to 2 vs. 1.1 for goingfrom 4 to 5 antennas). This delta increase is similar in magnitude tothe slope of the BER curve at large SNR.

Formally, a parameter called Diversity order d, is defined as the slopeof the BER curve as a function of SNR in the region of high SNR.

$d = {- {\lim\limits_{{SNR}\rightarrow\infty}{\log\frac{{BER}({SNR})}{SNR}}}}$

FIG. 18 shows the gains possible with MIMO as more receive antennas areadded. As more and more antennas are added, a Rayleigh channelapproaches the AWGN channel. Can one keep on increasing the number ofantennas indefinitely? No, beyond a certain number, increase in numberof paths L does not lead to significant gains. When complexity is takeninto account, a small number of antennas is enough for satisfactoryperformance.

Capacity of MIMO and SISO Channels

All system designs strive for a target capacity of throughput. For SISOchannels, the capacity is calculated using the well-known Shannonequation. Shannon defines capacity for an ergodic channel that data ratewhich can be transmitted with asymptotically small probability of error.The capacity of such a channel is given in terms of bits/sec or bynormalizing with bandwidth by bits/sec/Hz. The second formulation allowseasier comparison and is the one used more often. It is also bandwidthindependent.

$C = {W\;{\log_{2}\left( {1 + \frac{P}{N_{o}W}} \right)}b\text{/}s}$$\frac{C}{W} = {{\log_{2}\left( {1 + {SNR}} \right)}b\text{/}s\text{/}H_{z}}$

At high SNRs, ignoring the addition of 1 to SNR, the capacity is adirect function of SNR.

$\frac{C}{W} \approx {{\log_{2}({SNR})}b\text{/}s\text{/}H_{z}}$This capacity is based on a constant data rate and is not a function ofwhether channel state information is available to the receiver or thetransmitter. This result is applicable only to ergodic channels, oneswhere the data rate is fixed and SNR is stable.Capacity of MIMO Channels

Shannon's equation illustrates that a particular SNR can give only afixed maximum capacity. If SNR goes down, so will the ability of thechannel to pass data. In a fading channel, the SNR is constantlychanging. As the rate of fade changes, the capacity changes with it.

A fixed H matrix can be used as a benchmark of performance where thebasic assumption is that, for that one realization, the channel is fixedand hence has an ergodic channel capacity. In other words, for just thatlittle time period, the channel is behaving like an AWGN channel. Bybreaking a channel into portions of either time or frequency so that insmall segments, even in a frequency-selective channel with Doppler,channel can be treated as having a fixed realization of the H matrix,i.e. allowing us to think of it instantaneously as a AWGN channel. Thecapacity calculations are preformed over several realizations of Hmatrix and then compute average capacity over these. In flat fadingchannels the channel matrix may remain constant and or may change veryslowly. However, with user motion, this assumption does not hold.

Decomposing a MIMO Channel into Parallel Independent Channels

Conceptually MIMO may be thought of as the transmission of same dataover multiple antennas, hence it is a matrix channel. But there is amathematical trick that lets us decompose the MIMO channel into severalindependent parallel channels each of which can be thought of as a SISOchannel. To look at a MIMO channel as a set of independent channels, analgorithm called Singular Value Decomposition (SVD) is used. The processrequires pre-coding at the transmitter and receiver shaping at thereceiver.

Input and Output Auto-Correlation

Assuming that a MIMO channel has N transmitters and M receivers, thetransmitted vector across N_(T) antennas is given by x₁, x₂, x₃, . . .x_(NT). Individual transmit signals consist of symbols that are zeromean circular-symmetric complex Gaussian variables. (A vector x is saidto be circular-symmetric if e^(jθ)x has same distribution for all θ.)The covariance matrix for the transmitted symbols is written as:R _(xx) =E{xx ^(H)}Where symbol H stands for the transpose and component-wise complexconjugate of the matrix (also called Hermitian) and not the channelmatrix. This relationship gives us a measure of correlated-ness of thetransmitted signal amplitudes.

When the powers of the transmitted symbols are the same, a scaledidentity matrix is provided. For a (3×3) MIMO system of total power ofP_(T), equally distributed this matrix is written as:

$R_{xx} = {P_{T}\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}$

If the same system distributes the power differently say in ratio of1:2:3, then the covariance matrix would be:

$R_{xx} = {P_{T}\begin{bmatrix}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3\end{bmatrix}}$

If the total transmitted power is P_(T) and is equal to trace of theInput covariance matrix, the total power of the transmitted signal canbe written as the trace of the covariance matrix.P=tr{R _(xx)}The received signal is given by:r=Hx+n

The noise matrix (Nx1) components are assumed to be ZMGV (zero-meanGaussian variable) of equal variance. We can write the covariance matrixof the noise process similar to the transmit symbols as:R _(nn) =E{nn ^(H)}And since there is no correlation between its rows, this can be writtenas:R _(nn)=σ²1_(M)Which says that each of the M received noise signals is an independentsignal of noise variance, σ².

Each receiver receives a complex signal consisting of the sum of thereplicas from N transmit antennas and an independent noise signal.Assuming that the power received at each receiver is not the same, theSNR of the m_(th) receiver may be written as:

$\gamma_{m} = \frac{P_{m}}{\sigma^{2}}$where P_(m) is some part of the total power. However the average SNR forall receive antennas would still be equal to P_(T)/σ², where P_(T) istotal power because

$P_{T} = {\sum\limits_{m = 1}^{N_{T}}\; P_{m}}$Now we write the covariance matrix of the receive signal using as:R _(rr) =HR _(xx) H ^(H) +R _(nn)where R_(xx) is the covariance matrix of the transmitted signal. Thetotal receive power is equal to the trace of the matrix R_(rr).Singular Value Decomposition (SVD)

Referring now to FIG. 19 there is illustrated a modal decomposition of aMIMO channel with full CSI. SVD is a mathematical application that letsus create an alternate structure of the MIMO signal. In particular, theMIMO signal is examined by looking at the eigenvalues of the H matrix.The H matrix can be written in Singular Value Decomposition (SVD) formas:H=UΣV ^(H)where U and V are unitary matrices (U^(H)U=I_(NR), and V^(H)V=I_(NT))and Σ is a N_(R)×N_(T) diagonal matrix of singular values (σ_(i)) of Hmatrix. If H is a full Rank matrix then we have a min (N_(R),N_(T)) ofnon-zero singular values, hence the same number of independent channels.The parallel decomposition is essentially a linear mapping functionperformed by pre-coding the input signal by multiplying it with matrixV.x=V{tilde over (x)}

The received signal y is given by multiplying it with U^(H),{tilde over (y)}=U ^(H)(Hx+n)Now multiplying it out, and setting value of H, to get:{tilde over (y)}=U ^(H)(UΣV ^(H) x+n)Now substitute:x=V{tilde over (x)}To obtain:

$\begin{matrix}{\overset{\sim}{y} = {U^{H}\left( {{\underset{\_}{U\;\Sigma\; V^{H}}x} + n} \right)}} \\{= {U^{H}\left( {{U\;\Sigma\; V^{\dagger}V\overset{\sim}{x}} + n} \right)}} \\{= {{U^{H}U\;\Sigma\; V^{\dagger}V\overset{\sim}{x}} + {U^{H}n}}} \\{= {{\Sigma\;\overset{\sim}{x}} + \overset{\sim}{n}}}\end{matrix}$

In the last result, the output signal is in form of a pre-coded inputsignal times the singular value matrix, Σ. Note that the multiplicationof noise n, by the unitary matrix U^(H) does not change the noisedistribution. Note that the only way SVD can be used is if thetransmitter knows what pre-coding to apply, which of course requiresknowledge of the channel by the transmitter. As shown in FIG. 20, SVD isused for decomposing a matrix channel is decomposed into parallelequivalent channels.

Since SVD entails greater complexity, not the least of which is feedingback CSI to the transmitter, with the same results, why should weconsider the SVD approach? The answer is that the SVD approach allowsthe transmitter to optimize its distribution of transmitted power,thereby providing a further benefit of transmit array gain.

The channel eigenmodes (or principle components) can be viewed asindividual channels characterized by coefficients (eigenvalues). Thenumber of significant eigenvalues specifies the maximum degree ofdiversity. The larger a particular eigenvalue, the more reliable is thatchannel. The principle eigenvalue specifies the maximum possiblebeamforming gain. The most important benefit of the SVD approach is thatit allows for enhanced array gain—the transmitter can send more powerover the better channels, and less (or no) power over the worst ones.The number of principle components is a measure of the maximum degree ofdiversity that can be realized in this way.

Channel Capacity of a SIMO, MISO Channel

Before discussing the capacity of a MIMO channel, let's examine thecapacity of a channel that has multiple receivers or transmitters butnot both. We modify the SNR of a SISO channel by the gain factorobtained from having multiple receivers.C _(SIMO)=log₂(1+∥h∥ ² SNR)bits/s/HzThe channel consists of only N_(R) paths and hence the channel gain isconstrained by:∥h∥ ² =N _(R)This gives the ergodic capacity of the SIMO channel as:C _(SIMO)=log₂(1+N _(R) SNR)bits/s/HzSo the SNR is increasing by a factor of N_(R). This is a logarithmicgain. Note that we are assuming that the transmitter has no knowledge ofthe channel.

Next consider a MISO channel, with multiple transmitters but onereceiver. The channel capacity of a MISO channel is given by:

$C_{MISO} = {{\log_{2}\left( {1 + \frac{{h}^{2}{SNR}}{N_{T}}} \right)}{bits}\text{/}s\text{/}{Hz}}$

Why divide by N_(T)? Compared to the SIMO case, where each path has SNRbased on total power, in this case, total power is divided by the numberof transmitters. So the SNR at the one receiver keeps getting smaller asmore and more transmitters are added. For a two receiver case, each pathhas a half of the total power. But since there is only one receiver,this is being divided by the total noise power at the receiver, so theSNR is effectively cut in half.

Again if the transmitter has no knowledge of the channel, the equationdevolves in to a SISO channel:C _(MISO)=log₂(1+SNR)bits/s/HzThe capacity of a MISO channel is less than a SIMO channel when thechannel in unknown at the transmitter. However, if the channel is knownto the transmitter, then it can concentrate its power into one channeland the capacity of SIMO and MISO channel becomes equal under thiscondition.

Both SIMO and MISO can achieve diversity but they cannot achieve anymultiplexing gains. This is obvious for the case of one transmitter,(SIMO). In a MISO system all transmitters would need to send the samesymbol because a single receiver would have no way of separating thedifferent symbols from the multiple transmitters. The capacity stillincreases only logarithmically with each increase in the number of thetransmitters or the receivers. The capacity for the SIMO and MISO arethe same. Both channels experience array gain of the same amount butfall short of the MIMO gains.

Assuming a discrete MIMO channel model as shown in FIG. 21. The channelgain maybe time-varying but assumed to be fixed for a block of time andrandom. Assume that total transmit power is P, bandwidth is B and thePSD of noise process is N₀/2.

Assume that total power is limited by the relationship:

${E\left( {x^{H}x} \right)} = {{\sum\limits_{i = 1}^{N_{T}}\;{E\left\{ {x_{i}}^{2} \right\}}} = N_{T}}$The input covariance matrix may be written as R_(xx)R _(xx) =E{xx ^(H)}The trace of this matrix is equal to:

$\rho = {\frac{P}{\#{paths}} = {{tr}\left\{ R_{xx} \right\}}}$or the power per path. When the powers are uniformly distributed (equal)then this is equal to a unity matrix. The covariance matrix of theoutput signal would not be unity as it is a function of the H matrix.

Now the capacity expression for a MIMO matrix channel using a fixed butrandom realization the H matrix can be developed. Assuming availabilityof CSIR, the capacity of a deterministic channel is defined by Shannonas:

$\begin{matrix}{C = {\max\limits_{f\mspace{11mu}{(x)}}\;{I\left( {x;y} \right)}}}\end{matrix}$I(x;y) is called the mutual information of x and y. It is the capacityof the channel is the maximum information that can be transmitted from xto y by varying the channel PDF. The value f(x) is the probabilitydensity function of the transmit signal x. From information theory, therelationship of mutual information between two random variables as afunction of their differential entropy may be obtained.I(x;y)=

_((y))−

_((y|x))The second term is constant for a deterministic channel because it isfunction only of the noise. So mutual information is maximum only whenthe term H(y), called differential entropy is maximum.

The differential entropy H(y) is maximized when both x and y arezero-mean, Circular-Symmetric Complex Gaussian (ZMCSCG) random variable.Also from information theory, the following relationships are provided:

_((y))=log₂ {det(πeR _(yy)}

_((y|x))=log₂ {det(πeN _(o) I _(N) _(R) }

Now we write the signal y as:y=√{square root over (γ)}Hx+zHere γ is instantaneous SNR. The auto-correlation of the output signal ywhich we need for (27.48) is given by

$R_{yy} = {{E\mspace{11mu}\left\{ {yy}^{H} \right\}} = {{E\left\{ {\left( {{\sqrt{\gamma}{Hx}} + z} \right)\left( {{\sqrt{\gamma}x^{H}H^{H}} + z^{H}} \right)} \right\}} = {{E\left\{ \left( {{\gamma\mspace{11mu}{Hxx}^{H}H^{H}} + {zz}^{H}} \right) \right\}} = {{\gamma\;{HE}\mspace{11mu}\left\{ {xx}^{H} \right\} H^{H}} + {E\left\{ {zz}^{H} \right\}}}}}}$$\begin{matrix}{R_{yy} = {{\gamma\;{HR}_{xx}H^{H}} + {N_{O}I_{N_{R}}}}}\end{matrix}$

From here we can write the expression for capacity as

$\begin{matrix}{C = {\log_{2}\det\mspace{11mu}\left( {I_{N_{R}} + {\frac{SNR}{N_{T}}{HH}^{H}}} \right)}}\end{matrix}$When CSIT is not available, we can assume equal power distribution amongthe transmitters, in which case R_(xx) is an identity matrix and theequation becomes

$\begin{matrix}{C = {\log_{2}\det\mspace{11mu}\left( {I_{N_{R}} + {\frac{SNR}{N_{T}}{HH}^{H}}} \right)}}\end{matrix}$This is the capacity equation for MIMO channels with equal power. Theoptimization of this expression depends on whether or not the CSI (Hmatrix) is known to the transmitter.

Now note that as the number of antennas increases, we get

$\begin{matrix}{{\lim\limits_{N\rightarrow\infty}{\frac{I}{M}{HH}^{H}}} = I_{N}}\end{matrix}$This means that as the number of paths goes to infinity, the power thatreaches each of the infinite number of receivers becomes equal and thechannel now approaches an AWGN channel.

This gives us an expression about the capacity limit of a N_(T)×N_(R)MIMO system.C=M log₂ det(I _(N) _(R) +SNR)where M is the minimum of N_(T) and N_(R), the number of the antennas.Thus, the capacity increases linearly with M, the minimum of (N_(T),N_(R)). If a system has (4, 6) antennas, the maximum diversity that canbe obtained is of order 4, the small number of the two systemparameters.Channel Known at Transmitter

The SVD results can be used to determine how to allocate powers acrossthe transmitters to get maximum capacity. By allocating the powernon-equally, the capacity can be increased. In general, channels withhigh SNR (high σ_(i)), should get more power than those with lower SNR.

There is a solution to the power allocation problem at the transmittercalled the water-filling algorithm. This solution is given by:

$\frac{P_{i}}{P} = \left\{ \begin{matrix}\left( {\frac{I}{\gamma_{o}} - \frac{I}{\gamma_{i}}} \right) & {\gamma_{i} > \gamma_{o}} \\0 & {\gamma_{i} \leq \gamma_{o}}\end{matrix} \right.$Where γ₀ is a threshold constant. Here γ_(i) is the SNR of the i_(th)channel.

When comparing the inverse of the threshold with the inverse of thechannel SNR and the inverse difference is less than the threshold, nopower is allocated to the i_(th) channel. If the difference is positivethen more power may be added to see if it helps the overall performance.

The capacity using the water-filling algorithm is given by:

$C = {\sum\limits_{i:{\gamma_{i} > \gamma_{o}}}{B\mspace{11mu}{\log_{2}\left( \frac{\gamma_{i}}{\gamma_{o}} \right)}}}$The thing about the water-filling algorithm is that it is much easier tocomprehend then is it to describe using equations. Think of it as a boatsinking in the water. Where would a person sit on the boat while waitingfor rescue, clearly the part that is sticking above the water, right?The analogous part to the boat above the surface are the channels thatcan overcome fading. Some of the channels reach the receiver with enoughSNR for decoding. So the data/power should go to these channels and notto the ones that are under water. So basically, power is allocated tothose channels that are strongest or above a pre-set threshold.Channel Capacity in Outage

The Rayleigh channels go through such extremes of SNR fades that theaverage SNR cannot be maintained from one time block to the next. Due tothis, they are unable to support a constant data rate. A Rayleighchannel can be characterized as a binary state channel; an ergodicchannel but with an outage probability. When it has a SNR that is abovea minimum threshold, it can be treated as ON and capacity can becalculated using the information-theoretic rate. But when the SNR isbelow the threshold, the capacity of the channel is zero. The channel issaid to be in outage.

Although ergodic capacity can be useful in characterizing a fast-fadingchannel, it is not very useful for slow-fading, where there can beoutages for significant time intervals. When there is an outage, thechannel is so poor that there is no scheme able to communicate reliablyat a certain fixed data rate.

The outage capacity is the capacity that is guaranteed with a certainlevel of reliability. The outage capacity is defined as the informationrate that is guaranteed for (100−p) % of the channel realizations. A 1%outage probability means that 99% of the time the channel is above athreshold of SNR and can transmit data. For real systems, outagecapacity is the most useful measure of throughput capability.

Question: which would have higher capacity, a system with 1% outage or10% outage? The high probability of outage means that the threshold canbe set lower, which also means that system will have higher capacity, ofcourse only while it is working which is 90% of the time.

The capacity equation of a Rayleigh channel with outage probability

, can be written as:C _(out=(1−P) _(out) _()B log 2)(1γ_(min))The probability of obtaining a minimum threshold value of the SNR,assuming it has a Rayleigh distribution, can be calculated. The capacityof channel under outage probability ε is given by:

$= {\log_{2}\mspace{11mu}\left( {1 + {{{SNR} \cdot {In}}\mspace{11mu}\left( \frac{1}{1 - ɛ} \right)}} \right)}$The Shannon's equation has been modified by the outage probability.Capacity Under a Correlated Channel

MIMO gains come from the independence of the channels. The developmentof ergodic capacity assumes that channels created by MIMO areindependent. But what happens if there is some correlation among thechannels which is what happens in reality due to reflectors located nearthe base station or the towers. Usually in cell phone systems, thetransmitters (on account on being located high on towers) are lesssubject to correlation than are the receivers (the cell phones).

The signal correlation, r, between two antennas located a distance dapart, transmitting at the same frequency, is given by zero order Besselfunction defined as:

$r = {J_{o}^{2}\left( \frac{2\pi\; d}{\lambda} \right)}$where J₀(x) is the zero-th order Bessel function. FIG. 22 shows thecorrelation coefficient r, plotted between receive antennas vs. d/λusing the Jakes model.

An antenna that is approximately half a wavelength away experiences only10% correlation with the first. To examine the effect that correlationhas on system capacity, the channel mat channel matrix H is replaced inthe ergodic equation as follows:

$C = {\log_{2}\det\mspace{11mu}\left( {I_{N_{R}} + {\frac{SNR}{N_{T}}{HH}^{H}}} \right)}$

Assuming equal transmit powers, with a correlation matrix, assuming thatthe following normalization holds. This normalization allows thecorrelation matrix, rather than covariance.

${\sum\limits_{i,\;{j = 1}}^{N_{T},N_{R}}{h_{i,j}}^{2}} = 1$Now we write the capacity equation instead as

$C = {\log_{2}M\;\det\mspace{11mu}\left( {I + {\frac{SNR}{M} \cdot R}} \right)}$where R is the normalized correlation matrix, such that its components

${r_{{ij} = \frac{1}{\sqrt{\sigma_{i}\sigma_{j}}}}{\sum\limits_{K}{h_{ik}{h_{jk}}^{*}}}} = {\sum\limits_{K}{h_{ik}{h_{jk}}^{*}}}$

We can write the capacity equation as:C=M·log₂ det(I+SNR)+log₂ det(R)

The first underlined part of the expression is the capacity of Mindependent channels and the second is the contribution due tocorrelation. Since the determinant R is always <=1, then correlationalways results in degradation to the ergodic capacity.

An often used channel model for M=2, and 4 called the Kronecker Deltamodel takes this concept further by separating the correlation into twoparts, one near the transmitter and the other near the receiver. Themodel assumes each part to be independent of the other. Two correlationmatrices are defined, one for transmit, R_(T) and one for receiverR_(R). The complete channel correlation is assumed to be equal to theKronecker product of these two smaller matrices.R _(MIMO) =R _(R) ⊗R _(T)

The correlation among the columns of the H matrix represents thecorrelation between the transmitter and correlation between rows inreceivers. These two one-sided matrices can be written as:

$R_{R} = {\frac{1}{\beta}E\left\{ {HH}^{H} \right\}}$$R_{T} = {\frac{1}{\alpha}E\left\{ {H^{H}H} \right\}^{T}}$

The constant parameters (the correlation coefficients for each side)satisfy the relationship:αβ=Tr(R _(MIMO))

Now to see how correlation at the two ends affects the capacity, therandom channel H matrix is multiplied with the two correlation matricesas follows. The matrices can be produced in a number of fashions. Insome cases, test data is available which can be used in the matrix, inothers, a generic form based on Bessel coefficients is used. If thecorrelation coefficient on each side as a parameter is used, eachcorrelation matrix can be written as:

$R_{R} = {{\begin{bmatrix}1 & \sigma & \sigma^{2} \\\sigma & 1 & \sigma \\\sigma^{2} & \sigma & 1\end{bmatrix}\mspace{14mu}{and}\mspace{14mu} R_{R}} = \begin{bmatrix}1 & \beta & \beta^{2} \\\beta & 1 & \beta \\\beta^{2} & \beta & 1\end{bmatrix}}$Now write the correlated channel matrix in a Cholesky form asH=√{square root over (R _(T))}H _(W)√{square root over (R _(T))}where H_(W) is the random H matrix, that is now subject to correlationeffects.

The correlation at the transmitter is mathematically seen as correlationbetween the columns of the H matrix and can be written as R_(T). Thecorrelation at the receiver is seen as the correlation between the rowsof the H matrix, R_(R). Clearly if the columns are similar, each antennais seeing a similar channel. When the received amplitudes are similar ateach receiver, correlation at the receiver is seen. The H matrix undercorrelation is ill conditioned, and small changes lead to large changesin the received signal, clearly not a helpful situation.

The capacity of a channel with correlation can be written as:

$C = {\log_{2}{\det\left( {1_{NR} + {\frac{SNR}{N_{T}}R_{r}^{1/2}H^{H}R_{t}^{H\;{1/2}}}} \right)}}$When N_(T)=N_(R) and SNR is high, this expression can be approximatedas:

$C = {{\log_{2}{\det\left( {1_{NR} + {\frac{SNR}{N_{T}}H_{u}H_{u}^{H}}} \right)}} + {\log_{2}{\det\left( R_{r} \right)}} + {\log_{2}{\det\left( R_{t} \right)}}}$The last two terms are always negative since det(R)≤0. That implies thatcorrelation leads to reduction in capacity in frequency selectivechannels as shown in FIG. 23.

Here it is assumed that the frequency response is flat for the durationof the single realization of the H matrix. In FIG. 24 shows a channel2402 that is not flat. Its response is changing with frequency. The Hmatrix now changes within each sub-frequency of the signal. Note thatthis is not time, but frequency. The H matrix is written as a supermatrix of sub-matrices for each frequency.

Assume we can characterize the channel in N frequency sub-bands. The Hmatrix can now be written as a [(N×N_(R)),(N×N_(T))] matrix. A [3×3] Hmatrix is subdivided into N frequency and is written as an [18×18]matrix, with [3×3] matrices on the diagonal.

Spatial Multiplexing and how it Works

Each of the links in a MIMO system is assumed to transmit the sameinformation. This is an implicit assumption of obtaining diversity gain.Multicasting provides diversity gain but no data rate improvement. Ifindependent information could be sent across the antennas, then there isan opportunity to increase the data rate as well as keep some diversitygain. The data rate improvement in a MIMO system is called SpatialMultiplexing Gain (SMG).

The data rate improvement is related to the number of pairs of theRCV/XMT (receive/transmit) antennas, and when these numbers are unequal,it is proportional to smaller of the two numbers, N_(T), N_(R). This iseasy to see; the system can only transmit as many different symbols asthere are transmit antennas. This number is limited by the number ofreceive antennas, if the number of receive antennas is less than thenumber of transmit antennas.

Spatial multiplexing means the ability to transmit higher bit rate whencompared to a system where we only get diversity gains because oftransmissions of the same symbol from each transmitter. Therefore:

$d = {- {\lim_{{SNR}\rightarrow\infty}{\log\frac{{BER}({SNR})}{SNR}\mspace{14mu}{Diversity}\mspace{14mu}{Gain}}}}$$s = {\lim_{{SNR}\rightarrow\infty}{\frac{{Data}\mspace{14mu}{{Rate}({SNR})}}{\log({SNR})}\mspace{14mu}{Spatial}\mspace{14mu}{Multiplexing}\mspace{14mu}{Gain}}}$Should the diversity gain or multiplexing gain or maybe a little of bothbe used? The answer is that a little bit of both may be used.

One way to increase the number of independent Eigen channels is to use aset of orthogonal modes. Such a system transmits multiplecoaxially-propagating spatial modes each carrying an independent datastream through a single aperture pair. Therefore, the total capacity ofthe communication system can be increased by a factor equal to thenumber of transmitted modes. An orthogonal spatial modal basis set thathas gained interest recently is orbital angular momentum (OAM). Anelectromagnetic beam with a helical wavefront carries an OAMcorresponding to lℏ per photon, where ℏ is the reduced Planck constantand l is an unbounded integer. Importantly, OAM modes with different lvalues are mutually orthogonal, which allows them to be efficiently(de)multiplexed with low inter-modal crosstalk, thereby avoiding the useof multiple-input multiple-output (MIMO) processing.

Another approach for simultaneously transmitting multiple independentdata streams is to use MIMO-based spatial multiplexing, for whichmultiple aperture elements are employed at transmitter/receiver. As awell-established technique in wireless systems, this approach couldprovide capacity gains relative to single aperture systems and increaselink robustness for point-to-point (P2P) communications. In such asystem, each data-carrying beam is received by multiple spatiallyseparated receivers and MIMO signal processing is critical for reducingthe crosstalk among channels and thus allows data recovery.

However, MIMO signal processing becomes more onerous for MIMO-basedspatial multiplexing as the number of aperture elements increases. Inaddition, for OAM multiplexed systems, the detection of high-order OAMmodes presents a challenge for the receiver because OAM beams withlarger l values diverge more during propagation. Therefore, theachievable number of data channels for each type of multiplexingtechnique might be limited, and achieving a larger number of channels byusing any one approach would be significantly more difficult. Similar tothe multiplexing in few-mode and multi-core fibers, these two forms ofspatial multiplexing might be compatible with each other. Thecombination of them by fully exploiting the advantages of eachtechnique, such that they complement each other, might enable a densespatial multiplexed FSO system.

Antenna Placements in MIMO

FIGS. 25 and 26 illustrate the placement of antennas in a MIMO system.FIG. 25 illustrates antennas Tx₀ through Tx_(n-1) and receivers Rx₀through Rx_(n-1). FIG. 26 illustrates the transmission paths betweentransmitters Tx₀ and Tx₁ and receiver Rx.

The vectors describing the antenna placements are given by:a _(n) ^(t) =nd _(t) sin(θ_(t))n _(x) +nd _(t) cos(θ_(t))n _(z)a _(m) ^(r)=[D+md _(r) sin(θ_(r))cos(ϕ_(r))]n _(x) +md _(r) cos(θ_(t))n_(z) +md _(r) sin(θ_(r))sin(ϕ_(r))n _(y)The Euclidean distance between the antennas is:

$\begin{matrix}{d_{nm} = {{a_{m}^{r} - a_{n}^{t}}}} \\{= \left\lbrack {\left( {D + {{md}_{r}{\sin\left( \theta_{r} \right)}{\cos\left( \phi_{r} \right)}} - {{nd}_{t}{\sin\left( \theta_{t} \right)}}} \right)^{2} + \left( {{md}_{r}\sin\;\left( \theta_{r} \right){\sin\left( \phi_{t} \right)}} \right)^{2} +} \right.} \\\left. \left( {{{md}_{r}{\cos\left( \theta_{r} \right)}} - {{nd}_{t}{\cos\left( \theta_{t} \right)}}} \right)^{2} \right\rbrack^{1/2}\end{matrix}$

Since distance D is much larger than the antenna spacing, then:

${d_{nm} \approx {D + {{md}_{r}\sin\;\theta_{r}\cos\;\phi_{r}} - {{nd}_{t}\sin\;\theta_{t}} + {\frac{1}{2\; D}\left\lbrack {\left( {{{md}_{r}\cos\;\theta_{r}} - {{nd}_{t}\cos\;\theta_{t}}} \right)^{2} + \left( {{md}_{r}\sin\;\theta_{r}\sin\;\phi_{r}} \right)^{2}} \right\rbrack}}} = {D + {{md}_{r}\sin\;\theta_{r}\cos\;\phi_{r}} - {{nd}_{t}\sin\;\theta_{t}} + {\frac{1}{2\; D}\left\lbrack {{m^{2}d_{r}^{2}\cos^{2}\theta_{r}} + {n^{2}d_{t}^{2}\cos^{2}\theta_{t}} - {2\;{mnd}_{t}d_{r}\cos\;\theta_{t}\cos\;\theta_{r}} + {m^{2}d_{r}^{2}\sin^{2}\theta_{r}\sin^{2}\phi_{r}}} \right\rbrack}}$

Now criteria for the optimal antenna separation can be found. This isacheived by maximizing the capacity as a function of antenna separation.That is to maximize the product of the eigenvalues.

$W = \left\{ \begin{matrix}{{HH}^{H},} & {N \leq M} \\{{H^{H}H},} & {N > M}\end{matrix} \right.$This is obtained if H has orthogonal rows for N≤M or orthogonal columnsfor N>M. Defining the rows of H_(LOS) as h_(n) the orthogonality betweenthem can be expressed as:

$\left. \begin{matrix}{\mspace{79mu}{\left\langle {h_{n},h_{i}} \right\rangle_{n \neq i} = {\sum\limits_{m = 0}^{M - 1}\;{\exp\left( {{jk}\left( {d_{nm} - d_{im}} \right)} \right)}}}} \\{= {\sum\limits_{m = 0}^{M - 1}\;{{\exp\left( {j\; 2\pi\frac{d_{t}d_{r}{\cos\left( \theta_{r} \right)}{\cos\left( \theta_{t} \right)}}{\lambda\; D}\left( {i - n} \right)m} \right)} \cdot}}} \\{\exp\left( {{jk}\left\lbrack {{\left( {i - n} \right)d_{t}{\sin\left( \theta_{t} \right)}} + {\frac{1}{2\; D}\left( {i - n} \right)^{2}d_{t}{\cos^{2}\left( \theta_{t} \right)}}} \right\rbrack} \right)} \\{= 0}\end{matrix}\Rightarrow\frac{\sin\left( {{kd}_{t}d_{r}{\cos\left( \theta_{r} \right)}{\cos\left( \theta_{t} \right)}\left( {i - n} \right){M/2}\; D} \right)}{\sin\left( {{kd}_{t}d_{r}{\cos\left( \theta_{r} \right)}{\cos\left( \theta_{t} \right)}{\left( {i - n} \right)/2}\; D} \right.} \right. = {\left. 0\Rightarrow{d_{t}d_{r}} \right. = {\frac{\lambda\; D}{M\;{\cos\left( \theta_{t} \right)}{\cos\left( \theta_{r} \right)}}K}}$where K is a positive odd number usually chosen to be 1 since that givesthe smallest optimal antenna separation. Doing a similar derivation forthe case N>M would give the same expression only with M instead of N.Defining V=min(M,N), the general expression for antenna separation is:

${d_{t}d_{r}} = {\frac{\lambda\; D}{V\;{\cos\left( \theta_{t} \right)}{\cos\left( \theta_{r} \right)}}K}$That is the separation increases by distance D and decreases asfrequency increases.

Defining a new parameter η as:

$\eta = \sqrt{\frac{d_{t}d_{r}}{\left( {d_{t}d_{r}} \right)_{opt}}}$To quantify the deviation from optimality. Choosing d_(t)=d_(r)=dThen this reduces to η=d/d_(opt)Then the condition number for 2×2 MIMO will be

$\kappa = \sqrt{\frac{2 + \left\lbrack {2 + {2\;{\cos\left( {\pi\eta}^{2} \right)}}} \right\rbrack^{1/2}}{2 - \left\lbrack {2 + {2\;{\cos\left( {\pi\eta}^{2} \right)}}} \right\rbrack^{1/2}}}$

Calculating the distance d₁ and d₂ we have:d ₁ ² =d _(t) ²/4+D ² −d _(t) D cos(ϕ)d ₂ ² =d _(t) ²/4+D ² −d _(t) D cos(ϕ)

The gain relative to single RX antenna can be expressed as:

${G\left( {\phi,\alpha} \right)} = {{\frac{1}{T_{sc}}{\int_{0}^{T_{sc}}{\left( {{\sin\left( {{\omega\; t} + {d_{1}k}} \right)} + {\sin\left( {{\omega\; t} + {d_{2}k} + \alpha} \right)}} \right)^{2}{dt}}}} = {{{\cos\left( {{d_{1}k} - {d_{2}k} - \alpha} \right)} + 1} = {{\cos\left( {{k\;\Delta\; d} + \alpha} \right)} + 1}}}$Where:Δ=d ₂ −d ₁.But:

$\left( {d_{2} - d_{1}} \right)^{2} = {\left( {d_{1}^{2} + d_{2}^{2} - {2\; d_{1}d_{2}}} \right) = {{{\frac{d_{t}^{2}}{2} + {2\; D^{2}} - \sqrt{\frac{d_{t}^{4}}{4} + {4\; D^{4}} + {2\; d_{t}^{2}D^{2}} - {4\; d_{t}^{2}D^{2}{\cos^{2}(\phi)}}}} \approx {\frac{d_{t}^{2}}{2} + {2\; D^{2}} - {2\; D^{2}} - \frac{{d_{t}^{4}/4} + {2\; d_{t}^{2}D^{2}} - {4\; d_{t}^{2}D^{2}{\cos^{2}(\phi)}}}{2\; D^{2}}}} = {{d_{t}^{2}{\cos^{2}(\phi)}} - \frac{d_{t}^{4}}{16\; D^{2}}}}}$For D>>d_(t) ²:Δd≈d _(t) cos(ϕ)Therefore:

${G\left( {\phi,\alpha} \right)} = {{{\cos\left( {{k\;\Delta\; d} + \alpha} \right)} + 1} \approx {{\cos\left( {{\frac{2\pi}{\lambda}d_{t}{\cos(\phi)}} + \alpha} \right)} + 1}}$Wideband MIMO Model

For a narrowband MIMO we had r=sH+n which can be expressed as:

${r_{m}\lbrack j\rbrack} = {\sum\limits_{n = 1}^{N}\;{h_{nm}{s_{n}\lbrack j\rbrack}}}$Where j is the discrete symbol timing.

Now a wideband channel will act as a filter so that the gains need bereplaced by filters as:

${r_{m}\lbrack j\rbrack} = {\left. {\sum\limits_{n = 1}^{N}\;{\sum\limits_{i = {- L_{t}}}^{L_{t}}\;{{h_{nm}\left\lbrack {i,j} \right\rbrack}{s_{n}\left\lbrack {j - 1} \right\rbrack}}}}\Leftrightarrow r_{m} \right. = {\sum\limits_{n = 1}^{N}\;{h_{nm}*s_{n}}}}$Transmitting L_(B) symbols (block length) the channel matrix can beexpressed as:

$\Psi_{nm} = \begin{bmatrix}{h_{nm}\lbrack 0\rbrack} & \ldots & {h_{nm}\left\lbrack L_{t} \right\rbrack} & 0 & \ldots & 0 \\{h_{nm}\lbrack 1\rbrack} & {h_{nm}\lbrack 0\rbrack} & \ldots & {h_{nm}\left\lbrack {- L_{t}} \right\rbrack} & \ddots & 0 \\\vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\0 & \ldots & 0 & {h_{nm}\left\lbrack L_{t} \right\rbrack} & \ldots & {h_{nm}\lbrack 0\rbrack}\end{bmatrix}^{T}$Or

$= \begin{bmatrix}\Psi_{11} & \ldots & \Psi_{1M} \\\vdots & \ddots & \vdots \\\Psi_{N\; 1} & \ldots & \Psi_{NM}\end{bmatrix}$Defining signal matrix as:s=[s ₁ , . . . ,s _(N)]s _(n)=[s _(n)[1], . . . ,s _(n)[L _(B)]]Then the noise free received signal can be expressed as:r=sψ

For frequency selective channel, the capacity becomes frequencydependent. For a SISO channel, the capacity becomes:C=∫ _(−∞) ^(∞) log₂(1+γ|H(f)|²)dfAnd therefore for a MIMO channel the capacity becomes

$C = {\sum\limits_{i = 1}^{\min{({N,M})}}{\int_{- \infty}^{\infty}{{\log_{2}\left( {1 + {{\gamma\lambda}_{i}(f)}} \right)}{df}}}}$Line of Sight MIMO Channel

In LOS-MIMO there is one dominating path so that the incoming phases aredominated by the geometry of the channel and not scattering. Other thanthe LOS-path, there can be secondary paths caused by atmosphericscintillation or a reflection off the ground. This results in afrequency selective channel. The common model to use is a plane earthmodel with one reflection from ground. The two paths have some delaydifference τ which is typically set to 6.3 nsec in Rummler's model. Theimpulse response of such a channel is:h(t)=δ(t)+bδ(t−τ)e ^(jϕ) ⇔H(s)=

{h(t)}=1+be ^(−τs+jϕ)that is the channel introduces periodic notches in frequency spectrum.If one of the notches happen to be within the bandwidth, the channelwill suffer from sever fading and ISI.Dual Polarized MIMO

For a 4×4 Dual polarized MIMO, one can achieve four fold capacitycompared to SISO. The narrowband channel matrix can be written as:

$H = {\begin{bmatrix}h_{{1\; V},{1\; V}} & h_{{1\; V},{1\; H}} & h_{{1\; V},{2\; V}} & h_{{1\; V},{2H}} \\h_{{1\; H},{1\; V}} & h_{{1\; H},{1\; H}} & h_{{1H},{2\; V}} & h_{{1\; H},{2\; H}} \\h_{{2V},{1\; V}} & h_{{2\; V},{1\; H}} & h_{{2\; V},{2\; V}} & h_{{2\; V},{2\; H}} \\h_{{2H},{1\; V}} & h_{{2\; H},{1\; H}} & h_{{2\; H},{2\; V}} & h_{{2H},{2\; H}}\end{bmatrix} = {\begin{bmatrix}H_{11} & H_{12} \\H_{21} & H_{22}\end{bmatrix} = {\quad\begin{bmatrix}{\sqrt{{1 - \alpha}\;}e^{{jkd}_{11}}} & {\sqrt{\alpha\;}e^{{jkd}_{11}}} & {\sqrt{{1 - \alpha}\;}e^{{jkd}_{12}}} & {\sqrt{1 - \alpha}\; e^{{jkd}_{12}}} \\{\alpha\; e^{{jkd}_{11}}} & {\sqrt{{1 - \alpha}\;}e^{{jkd}_{11}}} & {\sqrt{\alpha\;}e^{{jkd}_{12}}} & {\sqrt{1 - \alpha}\; e^{{jkd}_{12}}} \\{\sqrt{{1 - \alpha}\;}e^{{jkd}_{21}}} & {\sqrt{\alpha\;}e^{{jkd}_{21}}} & {\sqrt{{1 - \alpha}\;}e^{{jkd}_{22}}} & {\sqrt{\alpha\;}e^{{jkd}_{22}}} \\{\alpha\; e^{{jkd}_{21}}} & {\sqrt{{1 - \alpha}\;}e^{{jkd}_{21}}} & {\sqrt{\alpha\;}e^{{jkd}_{22}}} & {\sqrt{1 - \alpha}\; e^{{jkd}_{22}}}\end{bmatrix}}}}$Where:

$H = {{H_{LoS} \otimes W_{XPD}} = {\begin{bmatrix}e^{{jkd}_{11}} & e^{{jkd}_{12}} \\e^{{jkd}_{21}} & e^{{jkd}_{22}}\end{bmatrix} \otimes \begin{bmatrix}\sqrt{1 - \alpha} & \sqrt{\alpha} \\\sqrt{\alpha} & \sqrt{1 - \alpha}\end{bmatrix}}}$

Here α measures the ratio of the power for one polarization that istransferred to the other polarization. Then:

$\alpha = \frac{1}{{XPD} + 1}$And condition numbers can be calculated as:c ₁=(2−f _(a)(a))[1−cos(πη₇ ²/2)]c ₂=(2−f _(a)(a))[cos(πη₇ ²/2)+1]c ₃=(2−f _(a)(a))[cos(πη₇ ²/2)+1]c ₄=(2−f _(a)(a))[1−cos(πη₇ ²/2)]Where

$\kappa = \sqrt{\frac{\max\left( {c_{1},c_{2},c_{3},c_{4}} \right)}{\min\left( {c_{1},c_{2},c_{3},c_{4}} \right)}}$

For SISO, assuming Gray coding and square QAM-M, the BER can becalculated as

$P_{b} = {\left( {1 - \frac{1}{\sqrt{M}}} \right)\mspace{11mu} Q\mspace{11mu}\left( \sqrt{\frac{3k}{M - 1}\frac{E_{b}}{N_{0}}} \right)}$

Therefore, the BER for a MIMO system using SVD can be expressed as:

$P_{b} = {\frac{1}{R_{H}}{\sum\limits_{i = 1}^{R_{H}}{\left( {1 - \frac{1}{\sqrt{M}}} \right)\mspace{11mu} Q\mspace{11mu}\left( \sqrt{\frac{3k}{M - 1}\frac{E_{b}}{N_{0}}\sigma_{i}^{2}} \right)}}}$MLO/QLO

Referring now to FIG. 27, wherein there is illustrated two manners forincreasing spectral efficiency 2702 of a communications system. Theincrease may be brought about by signal processing techniques 2704 inthe modulation scheme or using multiple access techniques. Additionally,the spectral efficiency can be increase by creating new Eigen channels2706 within the electromagnetic propagation. These two techniques arecompletely independent of one another and innovations from one class canbe added to innovations from the second class. Therefore, thecombination of these techniques introduced a further innovation.

Spectral efficiency 2702 is the key driver of the business model of acommunications system. The spectral efficiency is defined in units ofbit/sec/hz and the higher the spectral efficiency, the better thebusiness model. This is because spectral efficiency can translate to agreater number of users, higher throughput, higher quality or some ofeach within a communications system.

Regarding techniques using signal processing techniques or multipleaccess techniques. These techniques include innovations such as TDMA,FDMA, CDMA, EVDO, GSM, WCDMA, HSPA and the most recent OFDM techniquesused in 4G WIMAX and LTE. Almost all of these techniques use decades-oldmodulation techniques based on sinusoidal Eigen functions called QAMmodulation. Within the second class of techniques involving the creationof new Eigen channels 2706, the innovations include diversity techniquesincluding space and polarization diversity as well as multipleinput/multiple output (MIMO) where uncorrelated radio paths createindependent Eigen channels and propagation of electromagnetic waves.

The present communication system configuration introduces techniques,one from the signal processing techniques 2704 category and one from thecreation of new eigen channels 2706 category that are entirelyindependent from each other. Their combination provides a unique mannerto disrupt the access part of an end to end communications system fromtwisted pair and cable to fiber optics, to free space optics, to RF usedin cellular, backhaul and satellite, to RF satellite, to RF broadcast,to RF point-to point, to RF point-to-multipoint, to RF point-to-point(backhaul), to RF point-to-point (fronthaul to provide higher throughputCPRI interface for cloudification and virtualization of RAN andcloudified HetNet), to Internet of Things (IOT), to Wi-Fi, to Bluetooth,to a personal device cable replacement, to an RF and FSO hybrid system,to Radar, to electromagnetic tags and to all types of wireless access.One technique involves the use of a new signal processing techniqueusing new orthogonal signals to upgrade QAM modulation using nonsinusoidal functions. This is referred to as quantum level overlay(QLO). Application of the quantum level overlay (also called multiplelayer overlay) techniques uniquely offers orders of magnitude higherspectral efficiency within communication systems in their combination.

With respect to the quantum level overlay technique, new eigen functionsare introduced that when overlapped (on top of one another within asymbol) significantly increases the spectral efficiency of the systemand limit cross channel interference. The quantum level overlaytechnique borrows from quantum mechanics, special orthogonal signalsthat reduce the time bandwidth product and thereby increase the spectralefficiency of the channel while also limiting interference betweenchannels. Each orthogonal signal is overlaid within the symbol acts asan independent channel. These independent channels differentiate thetechnique from existing modulation techniques.

Multiple level overlay modulation techniques provide a new degree offreedom beyond the conventional 2 degrees of freedom, with time T andfrequency F being independent variables in a two-dimensional notationalspace defining orthogonal axes in an information diagram. This comprisesa more general approach rather than modeling signals as fixed in eitherthe frequency or time domain. Previous modeling methods using fixed timeor fixed frequency are considered to be more limiting cases of thegeneral approach of using multiple level overlay modulation. Within themultiple level overlay modulation technique, signals may bedifferentiated in two-dimensional space rather than along a single axis.Thus, the information-carrying capacity of a communications channel maybe determined by a number of signals which occupy different time andfrequency coordinates and may be differentiated in a notationaltwo-dimensional space to prevent channel interference.

Within the notational two-dimensional space, minimization of the timebandwidth product, i.e., the area occupied by a signal in that space,enables denser packing, and thus, the use of more signals, with higherresulting information-carrying capacity, within an allocated channel.Given the frequency channel delta (Δf), a given signal transmittedthrough it in minimum time Δt will have an envelope described by certaintime-bandwidth minimizing signals. The time-bandwidth products for thesesignals take the form:ΔtΔf=½(2n+1)where n is an integer ranging from 0 to infinity, denoting the order ofthe signal.

These signals form an orthogonal set of infinite elements, where eachhas a finite amount of energy. They are finite in both the time domainand the frequency domain, and can be detected from a mix of othersignals and noise through correlation, for example, by match filtering.Unlike other wavelets, these orthogonal signals have similar time andfrequency forms.

MLO/QLO may be differentiated from CDMA or OFDM by the manner in whichorthogonality among signals is achieved. MLO/QLO signals are mutuallyorthogonal in both time and frequency domains, and can be overlaid inthe same symbol time bandwidth product. Orthogonality is attained by thecorrelation properties, for example, by least sum of squares, of theoverlaid signals. In comparison, CDMA uses orthogonal interleaving ordisplacement of signals in the time domain, whereas OFDM uses orthogonaldisplacement of signals in the frequency domain.

Bandwidth efficiency may be increased for a channel by assigning thesame channel to multiple users. This is feasible if individual userinformation is mapped to special orthogonal functions. CDMA systemsoverlap multiple user information and views time intersymbol orthogonalcode sequences to distinguish pilot signals, and OFDM assigns uniquesignals to each pilot signal, but which are not overlaid, are onlyorthogonal in the frequency domain. Neither CDMA nor OFDM increasesbandwidth efficiency. CDMA uses more bandwidth than is necessary totransmit data when the signal has a low signal to noise ratio (SNR).OFDM spreads data over many subcarriers to achieve superior performancein multipath radiofrequency environments. OFDM uses a cyclic prefix OFDMto mitigate multipath effects and a guard time to minimize intersymbolinterference (ISI), and each channel is mechanistically made to behaveas if the transmitted waveform is orthogonal. (Sync function for eachsubcarrier in frequency domain.)

In contrast, MLO uses a set of functions which effectively form analphabet that provides more usable channels that do not interfere witheach other in the same bandwidth, thereby enabling high bandwidthefficiency. Some embodiments of MLO do not require the use of cyclicprefixes or guard times, and therefore, outperforms OFDM in spectralefficiency, peak to average power ratio, power consumption, and requiresfewer operations per bit. In addition, embodiments of MLO are moretolerant of amplifier nonlinearities than are CDMA and OFDM systems.

FIG. 28 illustrates an embodiment of an MLO/QLO transmitter system 2800,which receives input data stream 2799. The input data stream 2799comprises one or more pilot signals that are to be processed using theMLO/QLO techniques described herein to prevent pilot channelcontamination. The transceiver is realized using basic building blocks.System 2800 represents a modulator/controller, which is implemented on afield programmable gate array (FPGA) implementing a modulator/controller2801 that generates the MLO/QLO modulated pilot signals. In oneembodiment, the FPGA 2801 may be implemented using a Xlinx XUP Virtex IIPro Development System. The Virtex II Pro Development System is apowerful, versatile, low cost system providing multi-gigabittransceivers, system RAM, a system ACE compact flash controller, a fastEthernet interface and a USB programming interface. An example of anFPGA board is illustrated in FIG. 29. The FPGA 2801 implements most ofthe clock and data recovery as well as frequency estimations. The FPGAimplementation requires consideration of system requirements, standardsof consistency, usability, business objectives, minimization of latestage changes to product requirements and concepts, a reasonable designpath to be followed and contributory factors to be considered.

On the transmitter side, the FPGA board 2801 will generate the QLO Pilotsignals as well as the necessary control signals to enable thedigital-to-analog (DAC) boards 2804 and 2807. The block diagrams arerepresentative and do not include power amplifiers in the transmitter2808. The frequency band could be at any band, but here is the blockdiagram for 2.4-2.5 GHz (ISM band) where MIMO is currently being used.However, it should be understood that modulator/controller 2801 may useany quantity of pilot signals. Modulator/controller 2801 may alsocomprise an application specific integrated circuit (ASIC), and/or othercomponents, whether discrete circuit elements or integrated into asingle integrated circuit (IC) chip.

Modulator/controller 2801 is coupled to DACs 2804 and 2807,communicating a 10 bit I signal 2802 and a 10 bit Q signal 2805,respectively. In some embodiments, I signal 2802 and Q signal 2805correspond to pilot signals having QLO modulation applied thereto. Itshould be understood, however, that the 10 bit capacity of I signal 2802and Q signal 2805 is merely representative of an embodiment. Asillustrated, modulator/controller 2801 also controls DACs 2804 and 2807using control signals 2803 and 2806, respectively. In some embodiments,DACs 2804 and 2807 each comprise an AD5433, complementary metal oxidesemiconductor (CMOS) 10 bit current output DAC. In some embodiments,multiple control signals are sent to each of DACs 2804 and 2807.

DACs 2804 and 2807 output analogue signals 2815 and 2819 to quadraturemodulator 2821, which is coupled to LO 2820. The quadrature modulator2821 in one embodiment may comprise a direct conversion quad modulator(Analog AD8346) and the LO 2820 may comprise a Zcomm 2.4 GHz oscillator.The output of modulator 2820 is illustrated as coupled to a transmitter2808 to transmit the pilot signals wirelessly. In various embodiments,transmitter 2808 may also be coupled to a fiber-optic modem, a twistedpair, a coaxial cable, or other suitable transmission media ortransmission devices.

FIG. 30 illustrates an embodiment of an MLO/QLO receiver system 3000capable of receiving and demodulating pilot signals. System 3000receives an input pilot signal from a receiver 3008 that may compriseinput medium, such as RF, wired or optical. The modulator 3021 driven byLO 3020 converts the QLO/MLO modulated pilot signal input to baseband Isignal 3015 and Q signal 3019. I signal 3015 and Q signal 3019 are inputto analogue to digital converter (ADC) 3009.

ADC 3009 outputs 10 bit signal 3010 to demodulator/controller 3001 andreceives a control signal 3012 from demodulator/controller 3001.Demodulator/controller 3001 may comprise a field programmable gate array(FPGA) similar to that discussed herein above. In alternativeembodiments the demodulator/controller 3001 may also comprise anapplication specific integrated circuit (ASIC) and/or other components,whether discrete circuit elements or integrated into a single integratedcircuit (IC) chip. Demodulator/controller 3001 correlates received pilotsignals with locally generated replicas of the pilot signal set used, inorder to perform demodulation and identify the pilot signals sent.Demodulator/controller 3001 also estimates frequency errors and recoversthe data clock, which is used to read data from the ADC 3009. The clocktiming is sent back to ADC 3009 using control signal 3012, enabling ADC3009 to segment the digital I and Q signals 3017 and 3019 making up thetransmitted pilot symbols. In some embodiments, multiple control signalsare sent by demodulator/controller 3001 to ADC 3009.Demodulator/controller 3001 also outputs pilot signal 1301.

Hermite-Gaussian polynomials are a classical orthogonal polynomialsequence, which are the Eigenstates of a quantum harmonic oscillator.Signals based on Hermite-Gaussian polynomials possess the minimaltime-bandwidth product property described above, and may be used forembodiments of MLO/QLO systems. However, it should be understood thatother signals may also be used, for example orthogonal polynomials suchas Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials,Chebyshev polynomials, Laguerre-Gaussian polynomials, Hermite-Gaussianpolynomials and Ince-Gaussian polynomials. Q-functions are another classof functions that can be employed as a basis for MLO signals.

In quantum mechanics, a coherent state is a state of a quantum harmonicoscillator whose dynamics most closely resemble the oscillating behaviorof a classical harmonic oscillator system. A squeezed coherent state isany state of the quantum mechanical Hilbert space, such that theuncertainty principle is saturated. That is, the product of thecorresponding two operators takes on its minimum value. In embodimentsof an MLO/QLO system, operators correspond to time and frequency domainswherein the time-bandwidth product of the signals is minimized. Thesqueezing property of the signals allows scaling in time and frequencydomain simultaneously, without losing mutual orthogonality among thesignals in each layer. This property enables flexible implementations ofMLO systems in various communications systems.

Because signals with different orders are mutually orthogonal, they canbe overlaid to increase the spectral efficiency of a communicationchannel. For example, when n=0, the optimal baseband signal will have atime-bandwidth product of ½, which is the Nyquist Inter-SymbolInterference (ISI) criteria for avoiding ISI. However, signals withtime-bandwidth products of 3/2, 5/2, 7/2, and higher, can be overlaid toincrease spectral efficiency.

An embodiment of an MLO/QLO system uses functions based on modifiedHermite polynomials. The MLO/QLO pilot signals to be implemented on theFPGA 2801 are implemented in one embodiment according to the followingequations and are defined by:

${\psi_{n}\left( {t,\xi} \right)} = {\frac{\left( {\tanh\mspace{11mu}\xi} \right)^{n/2}}{2^{n/2}\left( {{n!}\cosh\mspace{11mu}\xi} \right)^{1/2}}e^{\frac{1}{2}{t^{2}{\lbrack{1 - {\tanh\;\xi}}\rbrack}}}{H_{n}\left( \frac{t}{\sqrt{\cosh\mspace{11mu}\xi\mspace{11mu}\sinh\mspace{11mu}\xi}} \right)}}$where t is time, and ξ is a bandwidth utilization parameter comprising aconstant (squeezing parameter to peg the signal bandwidth. Plots ofΨ_(n) for n ranging from 0 to 9, along with their Fourier transforms(amplitude squared), are shown in FIGS. 31A-31K. The orthogonality ofdifferent orders of the functions may be verified by integrating:∫∫ψ_(n)(t,ξ)ψ_(m)(t,ξ)dtdξ

The Hermite polynomial is defined by the contour integral:

${H_{n}(z)} = {\frac{n!}{2{\pi!}}{\oint{e^{{- t^{2}} + {2t\; 2}}t^{{- n} - 1}{dt}}}}$where the contour encloses the origin and is traversed in acounterclockwise direction. Hermite polynomials are described inMathematical Methods for Physicists, by George Arfken, for example onpage 416, the disclosure of which is incorporated by reference.

The first few Hermite polynomials are

$\begin{matrix}\begin{matrix}{{H_{0}(x)} = 1} \\{{H_{1}(x)} = {2x}} \\{{H_{2}(x)} = {{4x^{2}} - 2}} \\{{H_{3}(x)} = {{8x^{3}} - {12x}}} \\{{H_{4}(x)} = {{16x^{4}} - {48x^{2}} + 12}} \\{{H_{5}(x)} = {{32x^{5}} - {160x^{3}} + {120x}}} \\{{H_{6}(x)} = {{64x^{6}} - {480x^{4}} + {720x^{2}} - 120}} \\{{H_{7}(x)} = {{128x^{7}} - {1344x^{5}} + {3360x^{3}} - {1680x}}} \\{{H_{8}(x)} = {{256x^{8}} = {{3584x^{6}} + {13440x^{4}} - {13440x^{2}} + 1680}}} \\{{H_{9}(x)} = {{512x^{9}} - {9216x^{7}} + {48384x^{5}} - {80640x^{3}} + {30240x}}} \\{{H_{10}(x)} = {{1024x^{10}} - {23040x^{8}} + {161280x^{6}} - {403200x^{4}} + {302400x^{2}} - 30240.}}\end{matrix} & \; \\{{H_{n + 1}(x)} = {{2{{xH}_{n}(x)}} - {2{{nH}_{n - 1}(x)}}}} & \; \\{{H_{2k}(x)} = {\left( {- 1} \right)^{k}2^{k}{\left( {{2k} - 1} \right)!!} \times \left\lbrack {1 + {\sum\limits_{j = 1}^{k}{\frac{\left( {{- 4}k} \right)\left( {{{- 4}k} + 4} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {{{- 4}k} + {4j} - 4} \right)}{\left( {2j} \right)!}{x^{2j}.}}}} \right.}} & \; \\{{H_{{2k} + 1}(x)} = {\left( {- 1} \right)^{k}2^{k + 1}{\left( {{2k} + 1} \right)!!} \times \left\lbrack {x + {\sum\limits_{j = 1}^{k}{\frac{\left( {{- 4}k} \right)\left( {{{- 4}k} + 4} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {{{- 4}k} + {4j} - 4} \right)}{\left( {{2j} + 1} \right)!}{x^{2}.}}}} \right.}} & \;\end{matrix}$

FIGS. 31A-31K illustrate representative MLO signals and their respectivespectral power densities based on the modified Hermite polynomials Ψnfor n ranging from 0 to 9. FIG. 31A shows plots 3101 and 3104. Plot 3101comprises a curve 3127 representing TO plotted against a time axis 3102and an amplitude axis 3103. As can be seen in plot 3101, curve 3127approximates a Gaussian curve. Plot 3104 comprises a curve 3137representing the power spectrum of ψ 0 plotted against a frequency axis3105 and a power axis 3106. As can be seen in plot 3104, curve 3137 alsoapproximates a Gaussian curve. Frequency domain curve 3107 is generatedusing a Fourier transform of time domain curve 3127. The units of timeand frequency on axis 3102 and 3105 are normalized for basebandanalysis, although it should be understood that since the time andfrequency units are related by the Fourier transform, a desired time orfrequency span in one domain dictates the units of the correspondingcurve in the other domain. For example, various embodiments of MLO/QLOsystems may communicate using symbol rates in the megahertz (MHz) orgigahertz (GHz) ranges and the non-0 duration of a symbol represented bycurve 3127, i.e., the time period at which curve 3127 is above 0 wouldbe compressed to the appropriate length calculated using the inverse ofthe desired symbol rate. For an available bandwidth in the megahertzrange, the non-0 duration of a time domain signal will be in themicrosecond range.

FIGS. 31B-31J show plots 3107-3124, with time domain curves 3128-3136representing Ψ1 through Ψ9, respectively, and their correspondingfrequency domain curves 3138-3146. As can be seen in FIGS. 31A-31J, thenumber of peaks in the time domain plots, whether positive or negative,corresponds to the number of peaks in the corresponding frequency domainplot. For example, in plot 3123 of FIG. 31J, time domain curve 3136 hasfive positive and five negative peaks. In corresponding plot 3124therefore, frequency domain curve 3146 has ten peaks.

FIG. 31K shows overlay plots 3125 and 3126, which overlay curves3127-3136 and 3137-3146, respectively. As indicated in plot 3125, thevarious time domain curves have different durations. However, in someembodiments, the non-zero durations of the time domain curves are ofsimilar lengths. For an MLO system, the number of signals usedrepresents the number of overlays and the improvement in spectralefficiency. It should be understood that, while ten signals aredisclosed in FIGS. 31A-31K, a greater or lesser quantity of signals maybe used, and that further, a different set of signals, rather than thegin signals plotted, may be used.

MLO signals used in a modulation layer have minimum time-bandwidthproducts, which enable improvements in spectral efficiency, limitchannel interference, and are quadratically integrable. This isaccomplished by overlaying multiple demultiplexed parallel data streams,transmitting them simultaneously within the same bandwidth. The key tosuccessful separation of the overlaid data streams at the receiver isthat the signals used within each symbols period are mutuallyorthogonal. MLO overlays orthogonal signals within a single symbolperiod. This orthogonality prevents inter-symbol interference (ISI) andinter-carrier interference (ICI).

Because MLO works in the baseband layer of signal processing, and someembodiments use QAM architecture, conventional wireless techniques foroptimizing air interface, or wireless segments, to other layers of theprotocol stack will also work with MLO. Techniques such as channeldiversity, equalization, error correction coding, spread spectrum,interleaving and space-time encoding are applicable to MLO. For example,time diversity using a multipath-mitigating rake receiver can also beused with MLO. MLO provides an alternative for higher order QAM, whenchannel conditions are only suitable for low order QAM, such as infading channels. MLO can also be used with CDMA to extend the number oforthogonal channels by overcoming the Walsh code limitation of CDMA. MLOcan also be applied to each tone in an OFDM signal to increase thespectral efficiency of the OFDM systems.

Embodiments of MLO systems amplitude modulate a symbol envelope tocreate sub-envelopes, rather than sub-carriers. For data encoding, eachsub-envelope is independently modulated, resulting in each sub-envelopeindependently carrying information. Rather than spreading informationover many sub-carriers, for MLO, each sub-envelope of the carriercarries separate information. This information can be recovered due tothe orthogonality of the sub-envelopes defined with respect to the sumof squares over their duration and/or spectrum. Pulse trainsynchronization or temporal code synchronization, as needed for CDMA, isnot an issue, because MLO is transparent beyond the symbol level. MLOaddresses modification of the symbol, but since CDMA and TDMA arespreading techniques of multiple symbol sequences over time. MLO can beused along with CDMA and TDMA.

The orthogonality of PSI functions with different numbers n may bechecked by evaluating the double integral:∫∫ψ_(n)(β,ξ)ψ_(m)(β,ξ)dβdξ.

For computational purposes limits of integration should be taken torestrict area where ψ_(n)(β,ξ) is not equal to zero. We know fromtheoretical speculations that these functions are orthogonal if n iseven and m is odd or otherwise. Also overlap of two psi functions shouldbe very small if their numbers n and m are distinct.

Since each of the layers within the MLO signal comprises a differentchannel, different pilot channels may share a same bandwidth by beingassigned to different MLO layers within a same bandwidth. Thus, within asame bandwidth, pilot channel one may be assigned to a first MLO layer,pilot channel two may be assigned to a second MLO layer and so forth.

Referring now to FIGS. 32 and 33, there are more particularlyillustrated the transmit subsystem (FIG. 32) and the receiver subsystem(FIG. 33). The transceiver is realized using basic building blocksavailable as Commercially-Off-The-Shelf products. Modulation,demodulation and Special Hermite correlation and de-correlation areimplemented on a FPGA board. The FPGA board 3302 at the receiver 3300estimated the frequency error and recovers the data clock (as well asdata), which is used to read data from the analog-to-digital (ADC) board3306. The FGBA board 3300 also segments the digital I and Q channels.

On the transmitter side 3200, the FPGA board 3202 realizes the specialhermite correlated pilot signal as well as the necessary control signalsto control the digital-to-analog (DAC) boards 3204 to produce analog I&Qbaseband channels for the subsequent up conversion within the directconversion quad modulator 3206. The direct conversion quad modulator3206 receives an oscillator signal from oscillator 3208. The ADC 3306receives the I&Q signals from the quad demodulator 3308 that receives anoscillator signal from 3310. A power amplifier in the transmitter and anLNA in the receiver may be used to amplify the communications pilotsignal. The frequency band of 2.4-2.5 GHz (ISM band) is selected, butany frequency band of interest may be utilized.

MIMO uses diversity to achieve some incremental spectral efficiency.Each of the signals from the antennas acts as an independent orthogonalchannel. With QLO, the gain in spectral efficiency comes from within thesymbol and each QLO signal acts as independent channels as they are allorthogonal to one another in any permutation. However, since QLO isimplemented at the bottom of the protocol stack (physical layer), anytechnologies at higher levels of the protocol (i.e. Transport) will workwith QLO. Therefore one can use all the conventional techniques withQLO. This includes RAKE receivers and equalizers to combat fading,cyclical prefix insertion to combat time dispersion and all othertechniques using beam forming and MIMO to increase spectral efficiencyeven further.

When considering spectral efficiency of a practical wirelesscommunication system, due to possibly different practical bandwidthdefinitions (and also not strictly bandlimited nature of actual transmitsignal), the following approach would be more appropriate.

Referring now to FIG. 34, consider the equivalent discrete time system,and obtain the Shannon capacity for that system (denoted by Cd).Regarding the discrete time system, for example, for conventional QAMsystems in AWGN, the system will be:y[n]=ax[n]+w[n]where a is a scalar representing channel gain and amplitude scaling,x[n] is the input signal (pilot signal symbol) with unit average energy(scaling is embedded in a), y[n] is the demodulator (matched filter)output symbol, and index n is the discrete time index.

The corresponding Shannon capacity is:Cd=log 2(1+|a|2/σ2)where σ2 is the noise variance (in complex dimension) and |a|2/σ2 is theSNR of the discrete time system.

Second, compute the bandwidth W based on the adopted bandwidthdefinition (e.g., bandwidth defined by −40 dBc out of band power). Ifthe symbol duration corresponding to a sample in discrete time (or thetime required to transmit Cd bits) is T, then the spectral efficiencycan be obtained as:C/W=Cd/(TW)bps/Hz

In discrete time system in AWGN channels, using Turbo or similar codeswill give performance quite close to Shannon limit Cd. This performancein discrete time domain will be the same regardless of the pulse shapeused. For example, using either SRRC (square root raised cosine) pulseor a rectangle pulse gives the same Cd (or Cd/T). However, when weconsider continuous time practical systems, the bandwidths of SRRC andthe rectangle pulse will be different. For a typical practical bandwidthdefinition, the bandwidth for a SRRC pulse will be smaller than that forthe rectangle pulse and hence SRRC will give better spectral efficiency.In other words, in discrete time system in AWGN channels, there islittle room for improvement. However, in continuous time practicalsystems, there can be significant room for improvement in spectralefficiency.

Referring now to FIG. 35, there is illustrated a PSD plot (BLANK) ofMLO, modified MLO (MMLO) and square root raised cosine (SRRC). Theillustration, demonstrates the better localization property of MLO. Anadvantage of MLO is the bandwidth. FIG. 35 also illustrates theinterferences to adjacent channels will be much smaller for MLO. Thiswill provide additional advantages in managing, allocating or packagingspectral resources of several channels and systems, and furtherimprovement in overall spectral efficiency.

Referring now to FIG. 36 and FIG. 37, there is illustrated the manner inwhich pilot contamination is caused and may be improved by the describedsystem. FIG. 36 illustrates the various signals that that may betransmitted over different pilot channels from a transmitter to areceiver. Each signal 3602 is individual shown from SH 0 down to SHO 8that are centered on a center frequency f_(c). FIG. 37 shows theoverlapped absolute Fourier transforms for SH 0-SH 9. Due to MLO/QLOmodulation, the signals will not interfere with each other due to theorthogonal functions applied thereto.

The application of MLO/QLO modulation within a multiuser MIMO systemenables the creation of a number of orthogonal Eigen functions and basissets. The system provides a number of benefits including satisfyingShannon, Nyquist and Gabor equations, providing finite energy and power,providing finite period in the time-domain, providing a finite spectrumin the frequency-domain. The application of MLO/QLO to pilot signalsprovides quadratically integrable signal. The upper limit on the numberof pilot signals that may be transmitted using the technique depends onthe stability of the oscillator that is used. MLO/QLO offers betterspectral efficiency, peak-to-average power, power consumption andoperations per bit.

Derivation of the Signals Used in Modulation

To do that, it would be convenient to express signal amplitude s(t) in acomplex form close to quantum mechanical formalism. Therefore thecomplex signal can be represented as:

ψ(t) = s(t) + j σ(t) where  s(t) ≡ real  signalσ(t) = imaginary  signal  (quadrature)${\sigma(t)} = {{\frac{1}{\pi}{\int_{- \infty}^{\infty}{{s(\tau)}\frac{d\;\tau}{\tau - t}{s(t)}}}} = {{- \frac{1}{\pi}}{\int_{- \infty}^{\infty}{{\sigma(t)}\frac{d\;\tau}{\tau - t}}}}}$Where s(t) and σ(t) are Hilbert transforms of one another and since σ(t)is a quadrature of s(t), they have similar spectral components. That isif they were the amplitudes of sound waves, the ear could notdistinguish one form from the other.

Let us also define the Fourier transform pairs as follows:

${\psi(t)} = {\frac{1}{\pi}{\int_{- \infty}^{\infty}{{\varphi(f)}e^{j\;\omega\; t}{df}}}}$${\varphi(f)} = {\frac{1}{\pi}{\int_{- \infty}^{\infty}{{\psi(t)}e^{{- j}\;\omega\; t}{dt}}}}$ψ^(*)(t)ψ(t) = [s(t)]² + [σ(t)]² + … ≡ signal  power

Let's also normalize all moments to M0:

M₀ = ∫₀^(τ)s(t)dt M₀ = ∫₀^(τ)φ ^(*)φ  df

Then the moments are as follows:

$M_{0} = {\int\limits_{0}^{\tau}{{s(t)}{dt}}}$$M_{1} = {\int\limits_{0}^{\tau}{{{ts}(t)}{dt}}}$$M_{2} = {\int\limits_{0}^{\tau}{t^{2}{s(t)}{dt}}}$$M_{N - 1} = {\int\limits_{0}^{\tau}{t^{N - 1}{s(t)}{dt}}}$

In general, one can consider the signal s(t) be represented by apolynomial of order N, to fit closely to s(t) and use the coefficient ofthe polynomial as representation of data. This is equivalent tospecifying the polynomial in such a way that its first N “moments” M_(j)shall represent the data. That is, instead of the coefficient of thepolynomial, we can use the moments. Another method is to expand thesignal s(t) in terms of a set of N orthogonal functions φk(t), insteadof powers of time. Here, we can consider the data to be the coefficientsof the orthogonal expansion. One class of such orthogonal functions aresine and cosine functions (like in Fourier series).

Therefore we can now represent the above moments using the orthogonalfunction ψ with the following moments:

$\overset{\_}{t} = \frac{\int{{\psi^{*}(t)}t\;{\psi(t)}{dt}}}{\int{{\psi^{*}(t)}{\psi(t)}{dt}}}$$\overset{\_}{t^{2}} = \frac{\int{{\psi^{*}(t)}{t\;}^{2}{\psi(t)}{dt}}}{\int{{\psi^{*}(t)}{\psi(t)}{dt}}}$$\overset{\_}{t^{n}} = \frac{\int{{\psi^{*}(t)}t^{n}\;{\psi(t)}{dt}}}{\int{{\psi^{*}(t)}{\psi(t)}{dt}}}$

Similarly,

$\overset{\_}{f} = \frac{\int{{\varphi^{*}(f)}f\;{\varphi(f)}{df}}}{\int{{\varphi^{*}(f)}{\varphi(f)}{df}}}$$\overset{\_}{f^{2}} = \frac{\int{{\varphi^{*}(f)}f^{2}\;{\varphi(f)}{df}}}{\int{{\varphi^{*}(f)}{\varphi(f)}{df}}}$$\overset{\_}{f^{n}} = \frac{\int{{\varphi^{*}(f)}f^{n}\;{\varphi(f)}{df}}}{\int{{\varphi^{*}(f)}{\varphi(f)}{df}}}$

If we did not use complex signal, then:f=0

To represent the mean values from time to frequency domains, replace:

φ(f) → ψ(t) $\left. f\rightarrow{\frac{1}{2\pi}\frac{d}{dt}} \right.$

These are equivalent to somewhat mysterious rule in quantum mechanicswhere classical momentum becomes an operator:

$\left. P_{x}\rightarrow{\frac{h}{2\pi\; j}\frac{\partial\;}{\partial x}} \right.$

Therefore using the above substitutions, we have:

$\overset{\_}{f} = {\frac{\int{{\varphi^{*}(f)}f\;{\varphi(f)}{df}}}{\int{{\varphi^{*}(f)}{\varphi(f)}{df}}} = {\frac{\int{{\psi^{*}(t)}\left( \frac{1}{2\pi\; j} \right)\frac{d\;{\psi(t)}}{dt}{dt}}}{\int{{\psi^{*}(t)}\;{\psi(t)}{dt}}} = {\left( \frac{1}{2\pi\; j} \right)\frac{\int{\psi^{*}\frac{d\;\psi}{dt}{dt}}}{\int{\psi^{*}\psi\;{dt}}}}}}$

And:

$\overset{\_}{f^{2}} = {\frac{\int{{\varphi^{*}(f)}f^{2}\;{\varphi(f)}{df}}}{\int{{\varphi^{*}(f)}{\varphi(f)}{df}}} = {\frac{\int{{\psi^{*}\left( \frac{1}{2\pi\; j} \right)}^{2}\frac{d^{2}}{{dt}^{2}}\psi\;{dt}}}{\int{\psi^{*}\;\psi\;{dt}}} = {{- \left( \frac{1}{2\pi} \right)^{2}}\frac{\int{\psi^{*}\frac{d^{2}}{{dt}^{2}}\psi\;{dt}}}{\int{\psi^{*}\psi\;{dt}}}}}}$$\mspace{85mu}{\overset{\_}{t^{2}} = \frac{\int{\psi^{*}{t\;}^{2}\psi\;{dt}}}{\int{\psi^{*}\psi\;{dt}}}}$

We can now define an effective duration and effective bandwidth as:

${\Delta\; t} = {\sqrt{2\pi\;\overset{\_}{\left( {t - \overset{\_}{t}} \right)^{2}}} = {2{\pi \cdot {rms}}\mspace{14mu}{in}\mspace{14mu}{time}}}$${\Delta\; f} = {\sqrt{2\pi\;\overset{\_}{\left( {f - \overset{\_}{f}} \right)^{2}}} = {2{\pi \cdot {rms}}\mspace{14mu}{in}\mspace{14mu}{frequency}}}$

But we know that:

$\overset{\_}{\left( {t - \overset{\_}{t}} \right)^{2}} = {\overset{\_}{t^{2}} - \left( \overset{\_}{t} \right)^{2}}$$\overset{\_}{\left( {f - \overset{\_}{f}} \right)^{2}} = {\overset{\_}{f^{2}} - \left( \overset{\_}{f} \right)^{2}}$

We can simplify if we make the following substitutions:τ=t−tΨ(τ)=ψ(t)e ^(−jωτ)ω₀=ω=2π f=2πf ₀

We also know that:(Δt)²(Δf)²=(ΔtΔf)²

And therefore:

$\left( {\Delta\; t\;\Delta\; f} \right)^{2} = {{\frac{1}{4}\left\lbrack {4\frac{\int\;{{\Psi^{*}(\tau)}\tau^{2}{\Psi(\tau)}d\;\tau\mspace{11mu}{\int{\frac{d\;\Psi^{*}}{d\;\tau}\frac{d\;\Psi}{d\;\tau}d\mspace{11mu}\tau}}}}{\left( {\int{{\Psi^{*}(\tau)}{\psi(\tau)}d\;\tau}} \right)^{2}}} \right\rbrack} \geq \left( \frac{1}{4} \right)}$$\left( {\Delta\; t\;\Delta\; f} \right) \geq \left( \frac{1}{2} \right)$

Now instead of (Δt Δf)≥(½) we are interested to force the equality (ΔtΔf)=(½) and see what signals satisfy the equality. Given the fixedbandwidth Δf, the most efficient transmission is one that minimizes thetime-bandwidth product (Δt Δf)=(½) For a given bandwidth Δf, the signalthat minimizes the transmission in minimum time will be a Gaussianenvelope. However, we are often given not the effective bandwidth, butalways the total bandwidth f2−f1. Now, what is the signal shape whichcan be transmitted through this channel in the shortest effective timeand what is the effective duration?

$\left. {{\Delta\; t}==\frac{\frac{1}{\left( {2\pi} \right)^{2}}{\int\limits_{f_{1}}^{f_{2}}{\frac{d\;\varphi^{*}}{df}\frac{d\;\varphi}{df}}}}{\int\limits_{f_{1}}^{f_{2}}{\varphi^{*}\varphi\;{df}}}}\rightarrow\min \right.$Where φ(f) is zero outside the range f₂−f₁.

To do the minimization, we would use the calculus of variations(Lagrange's Multiplier technique). Note that the denominator is constantand therefore we only need to minimize the numerator as:

$\left. {\Delta\; t}\rightarrow\left. \min\rightarrow{\delta{\int\limits_{f_{1}}^{f_{2}}{\left( {{\frac{d\;\varphi^{*}}{df}\frac{d\;\varphi}{df}} + {\Lambda\;\varphi^{*}\varphi}} \right){df}}}} \right. \right. = 0$First  Trem $\begin{matrix}{{\delta{\int\limits_{f_{1}}^{f_{2}}{\frac{d\;\varphi^{*}}{df}\frac{d\;\varphi}{df}{df}}}} = {\int{\left( {{\frac{d\;\varphi^{*}}{df}\delta\frac{d\;\varphi}{df}} + {\frac{d\;\varphi}{df}\delta\frac{d\;\varphi^{*}}{df}}} \right){df}}}} \\{= {\int{\left( {{\frac{d\;\varphi^{*}}{df}\frac{d\;\delta\;\varphi}{df}} + {\frac{d\;\varphi}{df}\frac{d\;\delta\;\varphi^{*}}{df}}} \right){df}}}} \\{= {\left\lbrack {{\frac{d\;\varphi^{*}}{df}\delta\;\varphi} + {\frac{d\;\varphi}{df}{\delta\varphi}^{*}}} \right\rbrack_{f_{1}}^{f_{2}} -}} \\{\int{\left( {{\frac{{d\;}^{2}\varphi^{*}}{{df}^{2}}\delta\;\varphi} + {\frac{{d\;}^{2}\varphi}{{df}^{2}}\delta\;\varphi^{*}}} \right){df}}} \\{= {\int{\left( {{\frac{{d\;}^{2}\varphi^{*}}{{df}^{2}}\delta\;\varphi} + {\frac{d^{2}\;\varphi}{{df}^{2}}\delta\;\varphi^{*}}} \right){df}}}}\end{matrix}$ Second  Trem${\delta{\int\limits_{f_{1}}^{f_{2}}{\left( {\Lambda\;\varphi^{*}\varphi} \right){df}}}} = {\Lambda{\int\limits_{f_{1}}^{f_{2}}{\left( {{\varphi^{*}\delta\;\varphi} + {\varphi\;\delta\;\varphi^{*}}} \right){df}}}}$${{Both}\mspace{20mu}{Trem}} = {{\int{\left\lbrack {{\left( {\frac{d^{2}\varphi^{*}}{{df}^{\; 2}} + {\Lambda\;\varphi^{*}}} \right)\delta\;\varphi} + {\left( {\frac{d^{2}\varphi}{{df}^{\mspace{11mu} 2}} + {\Lambda\;\varphi}} \right)\delta\;\varphi^{*}}} \right\rbrack{df}}} = 0}$

This is only possible if and only if:

$\left( {\frac{d^{2}\varphi}{{df}^{\; 2}} + {\Lambda\;\varphi}} \right) = 0$

The solution to this is of the form

${\varphi(f)} = {\sin\; k\;{\pi\left( \frac{f - f_{1}}{f_{2} - f_{1}} \right)}}$

Now if we require that the wave vanishes at infinity, but still satisfythe minimum time-bandwidth product:(ΔtΔf)=(½)

Then we have the wave equation of a Harmonic Oscillator:

${\frac{d^{2}{\Psi(\tau)}}{d\;\tau^{2}} + {\left( {\lambda - {\alpha^{2}\tau^{2}}} \right){\Psi(\tau)}}} = 0$which vanishes at infinity only if:

λ = α(2n + 1)$\psi_{n} = {{e^{{- \frac{1}{2}}\omega^{2}\tau^{2}}\frac{d^{n}}{d\;\tau^{n}}e^{{- \alpha^{2}}\tau^{2}}} \propto {H_{n}(\tau)}}$Where Hn(τ) is the Hermite Gaussian functions and:(ΔtΔf)=½(2n+1)So Hermit functions Hn(τ) occupy information blocks of ½, 3/2, 5/2, . .. with ½ as the minimum information quanta.

Thus, in this manner the effects of pilot channel contamination may begreatly limited by applying MLO/QLO modulation to the pilot signalsbeing transmitted from user devices to a base station. The pilot signalswill not cause contamination between pilot channels since the MLOmodulation will cause the pilot signals to be overlapping and orthogonalto each other such that the pilot signals do not interfere with eachother.

It will be appreciated by those skilled in the art having the benefit ofthis disclosure that this system and method for communication usingorbital angular momentum with multiple layer overlay modulation providesimproved bandwidth and data transmission capability. It should beunderstood that the drawings and detailed description herein are to beregarded in an illustrative rather than a restrictive manner, and arenot intended to be limiting to the particular forms and examplesdisclosed. On the contrary, included are any further modifications,changes, rearrangements, substitutions, alternatives, design choices,and embodiments apparent to those of ordinary skill in the art, withoutdeparting from the spirit and scope hereof, as defined by the followingclaims. Thus, it is intended that the following claims be interpreted toembrace all such further modifications, changes, rearrangements,substitutions, alternatives, design choices, and embodiments.

What is claimed is:
 1. A method for transmitting a pilot signal,comprising: generating the pilot signal at a transmitting unit;modulating the pilot signal using quantum level overlay modulation toapply at least one orthogonal function to the pilot signal, wherein themodulated pilot signal is mutually orthogonal in both time and frequencydomains, further wherein the modulated pilot signal minimizestime-frequency resources; transmitting the modulated pilot signal fromthe transmitting unit to a receiving unit over a pilot channel; andwherein the at least one orthogonal function applied to the pilot signalsubstantially reduces pilot channel contamination on the pilot channelfrom other pilot channels.
 2. The method 1 further including the stepsof: receiving the modulated pilot signal at the receiving unit over thepilot channel; demodulating the modulated pilot signal using quantumlevel overlay demodulation to remove the at least one orthogonalfunction from the pilot signal; and outputting the demodulated pilotsignal.
 3. The method of claim 2, wherein the step of receiving furthercomprises receiving the modulated pilot signal using one of a pluralityof antennas of a MIMO transceiver.
 4. The method of claim 2 furtherincluding the step of generating channel state information responsive tothe demodulated pilot signal.
 5. The method of claim 1, wherein themodulation of the pilot signal provides a new orthogonal basis set tothe pilot signal.
 6. The method of claim 1, wherein the at least oneorthogonal function applied to the pilot signal substantially reducespilot channel contamination caused by hardware impairment andnon-reciprocal transceivers.
 7. The method of claim 1, wherein the atleast one orthogonal function applied to the pilot signal substantiallyreduces pilot channel contamination caused by frequency reuse betweencells.
 8. The method of claim 1, wherein the step of transmittingfurther includes transmitting the modulated pilot signal from thetransmitting unit to the receiving unit over a pilot channel usingcoordinated beamforming.
 9. The method of claim 1, wherein the pilotsignal is generated responsive to at least one of a time-multiplexedpilot scheme and a superimposed pilot scheme, further wherein the pilotsignal is transmitted in dedicated time slots in the time-multiplexedpilot scheme and the pilot signal is superimposed with data andtransmitted in all time slots in the superimposed pilot scheme.
 10. Themethod for transmitting pilot signals within a MIMO transmission system,comprising: generating a plurality of pilot signals at a plurality oftransmitting units; modulating each of the plurality of pilot signalsusing quantum level overlay modulation to apply at least one orthogonalfunction to the plurality of pilot signals, wherein the modulated pilotsignals are mutually orthogonal in both time and frequency domains,further wherein the modulated plurality of pilot signal minimizestime-frequency resources; transmitting the modulated pilot signals fromthe plurality of transmitting units to a MIMO receiving unit including aplurality of receiving antennas over a plurality of pilot channels, eachof the plurality of pilot channels interconnecting one of the pluralityof transmitting units to one of the plurality of receiving antennas ofthe MIMO receiving unit; and wherein the at least one orthogonalfunction applied to the plurality of pilot signals substantially reducespilot channel contamination between the plurality of pilot channels. 11.The method 10 further including the steps of: receiving the plurality ofmodulated pilot signals at the MIMO receiving unit over the plurality ofpilot channels; demodulating the modulated plurality of pilot signalsusing quantum level overlay demodulation to remove the at least oneorthogonal function from each of the plurality of pilot signals; andoutputting the demodulated plurality of pilot signals.
 12. The method ofclaim 11 further including the step of generating channel stateinformation for each of the plurality of pilot channels responsive tothe demodulated plurality of pilot signals.
 13. The method of claim 10,wherein the modulation of the plurality of pilot signals provides a neworthogonal basis set to each of the plurality of pilot signals.
 14. Themethod of claim 10, wherein the at least one orthogonal function appliedto the plurality of pilot signals substantially reduces pilot channelcontamination caused by hardware impairment and non-reciprocaltransceivers.
 15. The method of claim 10, wherein the at least oneorthogonal function applied to the plurality of pilot signalssubstantially reduces pilot channel contamination caused by frequencyreuse between cells.
 16. The method of claim 10, wherein the step oftransmitting further includes transmitting each of the plurality ofmodulated pilot signals from the plurality of transmitting units to theplurality of antennas and the MIMO receiving unit over the plurality ofpilot channels using coordinated beamforming.
 17. The method of claim10, wherein the plurality of pilot signals are generated responsive toat least one of a time-multiplexed pilot scheme and a superimposed pilotscheme, further wherein the plurality of pilot signals are transmittedin dedicated time slots in the time-multiplexed pilot scheme and theplurality of pilot signals are superimposed with data and transmitted inall time slots in the superimposed pilot scheme.
 18. The method of claim10, wherein a minimum number of the plurality of pilot signals comprisesa total number of the plurality of transmitting units.
 19. A system fortransmitting pilot signals, comprising: first signal processingcircuitry for generating a pilot signal at a transmitting unit; QLOmodulation circuitry for modulating the pilot signal using quantum leveloverlay modulation to apply at least one orthogonal function to thepilot signal, wherein the modulated pilot signal is mutually orthogonalin both time and frequency domains, further wherein the modulated pilotsignal minimizes time-frequency resources; a transceiver fortransmitting the modulated pilot signal from the transmitting unit overa pilot channel; and wherein the at least one orthogonal functionapplied to the pilot signal substantially reduces pilot channelcontamination on the pilot channel from other pilot channels.
 20. Thesystem of claim 19 further including a receiving unit for receiving themodulated pilot signal over the pilot channel, demodulating themodulated pilot signal using the quantum level overlay modulation toremove the at least one orthogonal function from the pilot signal andoutputting the demodulated pilot signal.
 21. The system of claim 20,wherein the receiving unit further comprises a MIMO receiver including aplurality of antennas for receiving the modulated pilot signal.
 22. Thesystem of claim 20, wherein the receiving unit further generates channelstate information responsive to the demodulated pilot signal.
 23. Thesystem of claim 19, wherein the modulation of the pilot signal providesa new orthogonal basis set to the pilot signal.
 24. The system of claim19, wherein the at least one orthogonal function applied to the pilotsignal substantially reduces pilot channel contamination caused byhardware impairment and non-reciprocal transceivers.
 25. The system ofclaim 19, wherein the at least one orthogonal function applied to thepilot signal substantially reduces pilot channel contamination caused byfrequency reuse between cells.
 26. The system of claim 19, wherein thetransceiver further transmits the modulated pilot signal from thetransceiver to the receiving unit over the pilot channel usingcoordinated beamforming.
 27. The system of claim 19, wherein the firstsignal processing circuitry generates the pilot signal responsive to atleast one of a time-multiplexed pilot scheme and a superimposed pilotscheme, further wherein the pilot signal is transmitted in dedicatedtime slots in the time-multiplexed pilot scheme and the pilot signal issuperimposed with data and transmitted in all time slots in thesuperimposed pilot scheme.